-
Notifications
You must be signed in to change notification settings - Fork 6
/
Rational.swift
863 lines (777 loc) · 29.5 KB
/
Rational.swift
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
//
// Rational.swift
// NumericAnnex
//
// Created by Xiaodi Wu on 4/15/17.
//
/// A type to represent a rational value.
///
/// - Note: `Ratio` is a type alias for `Rational<Int>`.
///
/// Create new instances of `Rational<T>` by using integer literals and the
/// division (`/`) operator. For example:
///
/// ```swift
/// let x = 1 / 3 as Ratio // `x` is of type `Rational<Int>`
/// let y = 2 as Ratio // `y` is of type `Rational<Int>`
/// let z: Ratio = 2 / 3 // `z` is also of type `Rational<Int>`
///
/// print(x + y + z) // Prints "3"
/// ```
///
/// You can create an unreduced fraction by using the initializer
/// `Rational<T>.init(numerator:denominator:)`. For example:
///
/// ```swift
/// let a = Ratio(numerator: 3, denominator: 3)
/// print(a) // Prints "3/3"
/// ```
///
/// All arithmetic operations on values in canonical form (i.e. reduced to
/// lowest terms) return results in canonical form. However, operations on
/// values not in canonical form may or may not return results that are
/// themselves in canonical form. The property `canonicalized` is the canonical
/// form of any value.
///
/// Additional Considerations
/// -------------------------
///
/// ### Special Values
///
/// `Rational<T>` does not prohibit zero as a denominator. Any instance with a
/// positive numerator and zero denominator represents (positive) infinity; any
/// instance with a negative numerator and zero denominator represents negative
/// infinity; and any instance with zero numerator and zero denominator
/// represents NaN ("not a number").
///
/// As with floating-point types, `Rational<T>.infinity` compares greater than
/// every finite value and negative infinity, and `-Rational<T>.infinity`
/// compares less than every finite value and positive infinity. Infinite values
/// of the same sign compare equal to each other.
///
/// As with floating-point types, `Rational<T>.nan` does not compare equal to
/// any other value, including another NaN. Use the property `isNaN` to test if
/// a value is NaN. `Rational<T>` arithmetic operations are intended to
/// propagate NaN in the same manner as analogous floating-point operations.
///
/// ### Numerical Limits
///
/// When a value of type `Rational<T>` is in canonical form, the sign of the
/// numerator is the sign of the value; that is, in canonical form, the sign of
/// the denominator is always positive. Therefore, `-1 / T.min` cannot be
/// represented as a value of type `Rational<T>` because `abs(T.min)` cannot be
/// represented as a value of type `T`.
///
/// To ensure that every representable value of type `Rational<T>` has a
/// representable magnitude and reciprocal of the same type, an overflow trap
/// occurs when the division (`/`) operator is used to create a value of type
/// `Rational<T>` with numerator `T.min`.
@_fixed_layout
public struct Rational<T : SignedInteger> : Codable
where T : Codable & _ExpressibleByBuiltinIntegerLiteral,
T.Magnitude : UnsignedInteger {
// ---------------------------------------------------------------------------
// MARK: Stored Properties
// ---------------------------------------------------------------------------
/// The numerator of the rational value.
public var numerator: T
/// The denominator of the rational value.
public var denominator: T
// ---------------------------------------------------------------------------
// MARK: Initializers
// ---------------------------------------------------------------------------
/// Creates a new value from the given numerator and denominator without
/// computing its canonical form (i.e., without reducing to lowest terms).
///
/// To create a value reduced to lowest terms, use the division (`/`)
/// operator. For example:
///
/// ```swift
/// let x = 3 / 3 as Rational<Int>
/// print(x) // Prints "1"
/// ```
///
/// - Parameters:
/// - numerator: The new value's numerator.
/// - denominator: The new value's denominator.
@_transparent // @_inlineable
public init(numerator: T, denominator: T) {
self.numerator = numerator
self.denominator = denominator
}
/// Creates a new rational value from the given binary integer.
///
/// If `source` or its magnitude is not representable as a numerator of type
/// `T`, a runtime error may occur.
///
/// - Parameters:
/// - source: A binary integer to convert to a rational value.
@_transparent // @_inlineable
public init<Source : BinaryInteger>(_ source: Source) {
let t = T(source)
// Ensure that `t.magnitude` is representable as a `T`.
_ = T(t.magnitude)
self.numerator = t
self.denominator = 1
}
/// Creates a new rational value from the given binary floating-point value.
///
/// If `source` or its magnitude is not representable exactly as a ratio of
/// two signed integers of type `T`, a runtime error may occur.
///
/// - Parameters:
/// - source: A binary floating-point value to convert to a rational value.
@_transparent // @_inlineable
public init<Source : BinaryFloatingPoint>(_ source: Source) {
if source.isNaN { self = .nan; return }
if source == .infinity { self = .infinity; return }
if source == -.infinity { self = -.infinity; return }
if source.isZero { self = 0; return }
let exponent = source.exponent
let significandWidth = source.significandWidth
let shift = Source.Exponent(significandWidth) - exponent
if shift <= 0 {
self.numerator = T(source)
self.denominator = 1
return
}
let numerator = T(
Source(
sign: source.sign,
exponent: exponent + shift,
significand: source.significand
)
)
let denominator = T(Source(sign: .plus, exponent: shift, significand: 1))
// Ensure that `numerator.magnitude` and `denominator.magnitude` are each
// representable as a `T`.
_ = T(numerator.magnitude)
_ = T(denominator.magnitude)
self.numerator = numerator
self.denominator = denominator
}
}
extension Rational where T : FixedWidthInteger {
// ---------------------------------------------------------------------------
// MARK: Initializers (Constrained)
// ---------------------------------------------------------------------------
/// Creates a new rational value from the given binary floating-point value,
/// if it can be represented exactly.
///
/// If `source` or its magnitude is not representable exactly as a ratio of
/// two signed integers of type `T`, the result is `nil`.
///
/// - Note: This initializer creates only instances of
/// `Rational<T> where T : FixedWidthInteger`.
///
/// - Parameters:
/// - source: A floating-point value to convert to a rational value.
@_transparent // @_inlineable
public init?<Source : BinaryFloatingPoint>(exactly source: Source) {
// TODO: Document this initializer.
if source.isNaN { self = .nan; return }
if source == .infinity { self = .infinity; return }
if source == -.infinity { self = -.infinity; return }
if source.isZero { self = 0; return } // Consider -0.0 to be exactly 0.
let exponent = source.exponent
let significandWidth = source.significandWidth
let shift = Source.Exponent(significandWidth) - exponent
let bitWidth = T.bitWidth
if shift <= 0 {
guard exponent + 1 < bitWidth else { return nil }
self.numerator = T(source)
self.denominator = 1
return
}
guard significandWidth + 1 < bitWidth && shift + 1 < bitWidth else {
return nil
}
self.numerator = T(
Source(
sign: source.sign,
exponent: exponent + shift,
significand: source.significand
)
)
self.denominator = T(Source(sign: .plus, exponent: shift, significand: 1))
}
}
extension Rational {
// ---------------------------------------------------------------------------
// MARK: Static Properties
// ---------------------------------------------------------------------------
/// Positive infinity.
///
/// Infinity compares greater than all finite numbers and equal to other
/// (positive) infinite values.
@_transparent // @_inlineable
public static var infinity: Rational {
return Rational(numerator: 1, denominator: 0)
}
/// A quiet NaN ("not a number").
///
/// A NaN compares not equal, not greater than, and not less than every value,
/// including itself. Passing a NaN to an operation generally results in NaN.
@_transparent // @_inlineable
public static var nan: Rational {
return Rational(numerator: 0, denominator: 0)
}
// ---------------------------------------------------------------------------
// MARK: Static Methods
// ---------------------------------------------------------------------------
/// Compares the (finite) magnitude of two finite values, returning -1 if
/// `lhs.magnitude` is less than `rhs.magnitude`, 0 if `lhs.magnitude` is
/// equal to `rhs.magnitude`, or 1 if `lhs.magnitude` is greater than
/// `rhs.magnitude`.
@_versioned
internal static func _compareFiniteMagnitude(
_ lhs: Rational, _ rhs: Rational
) -> Int {
let ldm = lhs.denominator.magnitude
let rdm = rhs.denominator.magnitude
let gcd = T.Magnitude.gcd(ldm, rdm)
let a = rdm / gcd * lhs.numerator.magnitude
let b = ldm / gcd * rhs.numerator.magnitude
return a == b ? 0 : (a < b ? -1 : 1)
// FIXME: Use full-width multiplication to avoid trapping on overflow
// where `T : FixedWidthInteger, T.Magnitude : FixedWidthInteger`.
/*
let a = (rdm / gcd).multipliedFullWidth(by: lhs.numerator.magnitude)
let b = (ldm / gcd).multipliedFullWidth(by: rhs.numerator.magnitude)
return a.high == b.high
? (a.low == b.low ? 0 : (a.low < b.low ? -1 : 1))
: (a.high < b.high ? -1 : 1)
*/
}
/// Returns the quotient obtained by dividing the first value by the second,
/// trapping in case of arithmetic overflow.
///
/// - Parameters:
/// - lhs: The value to divide.
/// - rhs: The value by which to divide `lhs`.
@_transparent // @_inlineable
public static func / (lhs: Rational, rhs: Rational) -> Rational {
return lhs * rhs.reciprocal()
}
/// Divides the left-hand side by the right-hand side and stores the quotient
/// in the left-hand side, trapping in case of arithmetic overflow.
///
/// - Parameters:
/// - lhs: The value to divide.
/// - rhs: The value by which to divide `lhs`.
@_transparent // @_inlineable
public static func /= (lhs: inout Rational, rhs: Rational) {
lhs = lhs * rhs.reciprocal()
}
// ---------------------------------------------------------------------------
// MARK: Computed Properties
// ---------------------------------------------------------------------------
/// The canonical representation of this value.
@_transparent // @_inlineable
public var canonical: Rational {
let nm = numerator.magnitude, dm = denominator.magnitude
// Note that if `T` is a signed fixed-width integer type, `gcd(nm, dm)`
// could be equal to `-T.min`, which is not representable as a `T`. This is
// why the following arithmetic is performed with values of type
// `T.Magnitude`.
let gcd = T.Magnitude.gcd(nm, dm)
guard gcd != 0 else { return self }
let n = sign == .plus ? T(nm / gcd) : -T(nm / gcd)
let d = T(dm / gcd)
return Rational(numerator: n, denominator: d)
}
/// A Boolean value indicating whether the instance's representation is in
/// canonical form.
@_transparent // @_inlineable
public var isCanonical: Bool {
if denominator > 0 {
return T.Magnitude.gcd(numerator.magnitude, denominator.magnitude) == 1
}
return denominator == 0 && numerator.magnitude <= 1
}
/// A Boolean value indicating whether the instance is finite.
///
/// All values other than NaN and infinity are considered finite.
@_transparent // @_inlineable
public var isFinite: Bool {
return denominator != 0
}
/// A Boolean value indicating whether the instance is infinite.
///
/// Note that `isFinite` and `isInfinite` do not form a dichotomy because NaN
/// is neither finite nor infinite.
@_transparent // @_inlineable
public var isInfinite: Bool {
return denominator == 0 && numerator != 0
}
/// A Boolean value indicating whether the instance is NaN ("not a number").
///
/// Because NaN is not equal to any value, including NaN, use this property
/// instead of the equal-to operator (`==`) or not-equal-to operator (`!=`) to
/// test whether a value is or is not NaN.
@_transparent // @_inlineable
public var isNaN: Bool {
return denominator == 0 && numerator == 0
}
/// A Boolean value indicating whether the instance is a proper fraction.
///
/// A fraction is proper if and only if the absolute value of the fraction is
/// less than 1.
@_transparent // @_inlineable
public var isProper: Bool {
return denominator != 0 && numerator / denominator == 0
}
/// A Boolean value indicating whether the instance is equal to zero.
@_transparent // @_inlineable
public var isZero: Bool {
return denominator != 0 && numerator == 0
}
/// The magnitude (absolute value) of this value.
@_transparent // @_inlineable
public var magnitude: Rational {
return sign == .minus ? -self : self
}
/// The mixed form representing this value.
///
/// If the value is not finite, the mixed form has zero as its whole part and
/// the value as its fractional part.
@_transparent // @_inlineable
public var mixed: (whole: T, fractional: Rational) {
if denominator == 0 { return (whole: 0, fractional: self) }
let t = numerator.quotientAndRemainder(dividingBy: denominator)
return (
whole: t.quotient,
fractional: Rational(numerator: t.remainder, denominator: denominator)
)
}
/// The sign of this value.
@_transparent // @_inlineable
public var sign: Sign {
return numerator == 0 || (denominator < 0) == (numerator < 0)
? .plus
: .minus
}
// ---------------------------------------------------------------------------
// MARK: Methods
// ---------------------------------------------------------------------------
/// Returns the reciprocal (multiplicative inverse) of this value.
@_transparent // @_inlineable
public func reciprocal() -> Rational {
return numerator < 0
? Rational(numerator: -denominator, denominator: -numerator)
: Rational(numerator: denominator, denominator: numerator)
}
/// Returns this value rounded to an integral value using the specified
/// rounding rule.
///
/// ```swift
/// let x = 7 / 2 as Rational<Int>
/// print(x.rounded()) // Prints "4"
/// print(x.rounded(.towardZero)) // Prints "3"
/// print(x.rounded(.up)) // Prints "4"
/// print(x.rounded(.down)) // Prints "3"
/// ```
///
/// See the `FloatingPointRoundingRule` enumeration for more information about
/// the available rounding rules.
///
/// - Parameters:
/// - rule: The rounding rule to use.
///
/// - SeeAlso: `round(_:)`, `RoundingRule`
@_transparent // @_inlineable
public func rounded(
_ rule: RoundingRule = .toNearestOrAwayFromZero
) -> Rational {
var t = self
t.round(rule)
return t
}
/// Rounds the value to an integral value using the specified rounding rule.
///
/// ```swift
/// var x = 7 / 2 as Rational<Int>
/// x.round() // x == 4
///
/// var x = 7 / 2 as Rational<Int>
/// x.round(.towardZero) // x == 3
///
/// var x = 7 / 2 as Rational<Int>
/// x.round(.up) // x == 4
///
/// var x = 7 / 2 as Rational<Int>
/// x.round(.down) // x == 3
/// ```
///
/// See the `FloatingPointRoundingRule` enumeration for more information about
/// the available rounding rules.
///
/// - Parameters:
/// - rule: The rounding rule to use.
///
/// - SeeAlso: `round(_:)`, `RoundingRule`
@_transparent // @_inlineable
public mutating func round(_ rule: RoundingRule = .toNearestOrAwayFromZero) {
if denominator == 0 { return }
let f: T
(numerator, f) = numerator.quotientAndRemainder(dividingBy: denominator)
// Rounding rules only come into play if the fractional part is non-zero.
if f != 0 {
switch rule {
case .toNearestOrAwayFromZero:
fallthrough
case .toNearestOrEven:
switch denominator.magnitude.quotientAndRemainder(
dividingBy: f.magnitude
) {
case (2, 0): // Tie.
if rule == .toNearestOrEven && numerator % 2 == 0 { break }
fallthrough
case (1, _): // Nearest is away from zero.
if f > 0 { numerator += 1 } else { numerator -= 1 }
default: // Nearest is toward zero.
break
}
case .up:
if f > 0 { numerator += 1 }
case .down:
if f < 0 { numerator -= 1 }
case .towardZero:
break
case .awayFromZero:
if f > 0 { numerator += 1 } else { numerator -= 1 }
}
}
denominator = 1
}
}
extension Rational : ExpressibleByIntegerLiteral {
// ---------------------------------------------------------------------------
// MARK: ExpressibleByIntegerLiteral
// ---------------------------------------------------------------------------
@_transparent // @_inlineable
public init(integerLiteral value: T) {
self.numerator = value
self.denominator = 1
}
}
extension Rational : CustomStringConvertible {
// ---------------------------------------------------------------------------
// MARK: CustomStringConvertible
// ---------------------------------------------------------------------------
@_transparent // @_inlineable
public var description: String {
if numerator == 0 { return denominator == 0 ? "nan" : "0" }
if denominator == 0 { return numerator < 0 ? "-inf" : "inf" }
return denominator == 1 ? "\(numerator)" : "\(numerator)/\(denominator)"
}
}
extension Rational : Equatable {
// ---------------------------------------------------------------------------
// MARK: Equatable
// ---------------------------------------------------------------------------
@_transparent // @_inlineable
public static func == (lhs: Rational, rhs: Rational) -> Bool {
if lhs.denominator == 0 {
if lhs.numerator == 0 { return false }
if lhs.numerator > 0 { return rhs.denominator == 0 && rhs.numerator > 0 }
return rhs.denominator == 0 && rhs.numerator < 0
}
if rhs.denominator == 0 { return false }
return lhs.sign == rhs.sign && _compareFiniteMagnitude(lhs, rhs) == 0
}
}
extension Rational : Hashable {
// ---------------------------------------------------------------------------
// MARK: Hashable
// ---------------------------------------------------------------------------
// @_transparent // @_inlineable
public var hashValue: Int {
let t = canonical
return _Hash._combine(t.numerator, t.denominator)
}
}
extension Rational : Comparable {
// ---------------------------------------------------------------------------
// MARK: Comparable
// ---------------------------------------------------------------------------
@_transparent // @_inlineable
public static func < (lhs: Rational, rhs: Rational) -> Bool {
if lhs.denominator == 0 {
if lhs.numerator >= 0 { return false }
return rhs.denominator != 0 || rhs.numerator > 0
}
if rhs.denominator == 0 { return rhs.numerator > 0 }
switch (lhs.sign, rhs.sign) {
case (.plus, .minus):
return false
case (.minus, .plus):
return true
case (.plus, .plus):
return _compareFiniteMagnitude(lhs, rhs) < 0
case (.minus, .minus):
return _compareFiniteMagnitude(lhs, rhs) > 0
}
}
@_transparent // @_inlineable
public static func > (lhs: Rational, rhs: Rational) -> Bool {
return rhs < lhs
}
@_transparent // @_inlineable
public static func <= (lhs: Rational, rhs: Rational) -> Bool {
if lhs.denominator == 0 {
if lhs.numerator == 0 { return false }
if lhs.numerator > 0 { return rhs.denominator == 0 && rhs.numerator > 0 }
return rhs.denominator != 0 || rhs.numerator != 0
}
if rhs.denominator == 0 { return rhs.numerator > 0 }
switch (lhs.sign, rhs.sign) {
case (.plus, .minus):
return false
case (.minus, .plus):
return true
case (.plus, .plus):
return _compareFiniteMagnitude(lhs, rhs) <= 0
case (.minus, .minus):
return _compareFiniteMagnitude(lhs, rhs) >= 0
}
}
@_transparent // @_inlineable
public static func >= (lhs: Rational, rhs: Rational) -> Bool {
return rhs <= lhs
}
}
extension Rational : Strideable {
// ---------------------------------------------------------------------------
// MARK: Strideable
// ---------------------------------------------------------------------------
@_transparent // @_inlineable
public func distance(to other: Rational) -> Rational {
return other - self
}
@_transparent // @_inlineable
public func advanced(by amount: Rational) -> Rational {
return self + amount
}
}
extension Rational : Numeric {
// ---------------------------------------------------------------------------
// MARK: Numeric
// ---------------------------------------------------------------------------
@_transparent // @_inlineable
public init?<U>(exactly source: U) where U : BinaryInteger {
guard let t = T(exactly: source) else { return nil }
// Ensure that `t.magnitude` is representable as a `T`.
guard let _ = T(exactly: t.magnitude) else { return nil }
self.numerator = t
self.denominator = 1
}
@_transparent // @_inlineable
public static func + (lhs: Rational, rhs: Rational) -> Rational {
if lhs.denominator == 0 {
if rhs.denominator != 0 || lhs.numerator == 0 { return lhs }
if lhs.numerator > 0 { return rhs.numerator < 0 ? .nan : rhs }
return rhs.numerator > 0 ? .nan : rhs
}
if rhs.denominator == 0 { return rhs }
let ldm = lhs.denominator.magnitude
let rdm = rhs.denominator.magnitude
let gcd = T.Magnitude.gcd(ldm, rdm)
let a = T(rdm / gcd * lhs.numerator.magnitude)
let b = T(ldm / gcd * rhs.numerator.magnitude)
let n = lhs.sign == .plus
? (rhs.sign == .plus ? a + b : a - b)
: (rhs.sign == .plus ? b - a : -a - b)
let d = T(ldm / gcd * rdm)
return Rational(numerator: n, denominator: d).canonical
}
@_transparent // @_inlineable
public static func += (lhs: inout Rational, rhs: Rational) {
lhs = lhs + rhs
}
@_transparent // @_inlineable
public static func - (lhs: Rational, rhs: Rational) -> Rational {
return lhs + (-rhs)
}
@_transparent // @_inlineable
public static func -= (lhs: inout Rational, rhs: Rational) {
lhs = lhs + (-rhs)
}
@_transparent // @_inlineable
public static func * (lhs: Rational, rhs: Rational) -> Rational {
if lhs.denominator == 0 {
if rhs.numerator == 0 { return .nan }
return rhs.sign == .plus ? lhs : -lhs
}
if rhs.denominator == 0 {
if lhs.numerator == 0 { return .nan }
return lhs.sign == .plus ? rhs : -rhs
}
let lnm = lhs.numerator.magnitude, ldm = lhs.denominator.magnitude
let rnm = rhs.numerator.magnitude, rdm = rhs.denominator.magnitude
// Note that if `T` is a signed fixed-width integer type, `gcd(lnm, rdm)` or
// `gcd(rnm, ldm)` could be equal to `-T.min`, which is not representable as
// a `T`. This is why the following arithmetic is performed with values of
// type `T.Magnitude`.
let a = T.Magnitude.gcd(lnm, rdm)
let b = T.Magnitude.gcd(rnm, ldm)
let n = lhs.sign == rhs.sign
? T(lnm / a * (rnm / b))
: -T(lnm / a * (rnm / b))
let d = T(ldm / b * (rdm / a))
return Rational(numerator: n, denominator: d)
}
@_transparent // @_inlineable
public static func *= (lhs: inout Rational, rhs: Rational) {
lhs = lhs * rhs
}
}
extension Rational : SignedNumeric {
// ---------------------------------------------------------------------------
// MARK: SignedNumeric
// ---------------------------------------------------------------------------
@_transparent // @_inlineable
public static prefix func - (operand: Rational) -> Rational {
return Rational(
numerator: -operand.numerator, denominator: operand.denominator
)
}
@_transparent // @_inlineable
public mutating func negate() {
numerator.negate()
}
}
/// Returns the absolute value (magnitude) of `x`.
@_transparent
public func abs<T>(_ x: Rational<T>) -> Rational<T> {
return x.magnitude
}
/// Returns the closest integral value greater than or equal to `x`.
@_transparent
public func ceil<T>(_ x: Rational<T>) -> Rational<T> {
return x.rounded(.up)
}
/// Returns the closest integral value less than or equal to `x`.
@_transparent
public func floor<T>(_ x: Rational<T>) -> Rational<T> {
return x.rounded(.down)
}
/// Returns the closest integral value; if two values are equally close, returns
/// the one with greater magnitude.
@_transparent
public func round<T>(_ x: Rational<T>) -> Rational<T> {
return x.rounded()
}
/// Returns the closest integral value with magnitude less than or equal to that
/// of `x`.
@_transparent
public func trunc<T>(_ x: Rational<T>) -> Rational<T> {
return x.rounded(.towardZero)
}
public typealias Ratio = Rational<Int>
// MARK: -
extension BinaryInteger {
// ---------------------------------------------------------------------------
// MARK: Initializers
// ---------------------------------------------------------------------------
/// Creates a new binary integer from the given rational value, if it can be
/// represented exactly.
///
/// If `source` is not representable exactly, the result is `nil`.
///
/// - Parameters:
/// - source: A rational value to convert to a binary integer.
@_transparent // @_inlineable
public init?<U>(exactly source: Rational<U>) {
let (whole, fraction) = source.mixed
guard fraction.isZero, let exact = Self(exactly: whole) else { return nil }
self = exact
}
/// Creates a new binary integer from the given rational value, rounding
/// toward zero.
///
/// If `source` is outside the bounds of this type after rounding toward zero,
/// a runtime error may occur.
///
/// - Parameters:
/// - source: A rational value to convert to a binary integer.
@_transparent // @_inlineable
public init<U>(_ source: Rational<U>) {
self = Self(source.mixed.whole)
}
}
extension FloatingPoint {
// ---------------------------------------------------------------------------
// MARK: Initializers
// ---------------------------------------------------------------------------
/// Creates a new floating-point value from the given rational value, after
/// rounding the whole part, the numerator of the fractional part, and the
/// denominator of the fractional part each to the closest possible
/// representation.
///
/// If two representable values are equally close, the result of rounding is
/// the value with more trailing zeros in its significand bit pattern.
///
/// - Parameters:
/// - source: The rational value to convert to a floating-point value.
public init(_ source: Rational<Int>) {
let (whole, fraction) = source.mixed
self = Self(whole) + Self(fraction.numerator) / Self(fraction.denominator)
}
/// Creates a new floating-point value from the given rational value, after
/// rounding the whole part, the numerator of the fractional part, and the
/// denominator of the fractional part each to the closest possible
/// representation.
///
/// If two representable values are equally close, the result of rounding is
/// the value with more trailing zeros in its significand bit pattern.
///
/// - Parameters:
/// - source: The rational value to convert to a floating-point value.
public init(_ source: Rational<Int8>) {
let (whole, fraction) = source.mixed
self = Self(whole) + Self(fraction.numerator) / Self(fraction.denominator)
}
/// Creates a new floating-point value from the given rational value, after
/// rounding the whole part, the numerator of the fractional part, and the
/// denominator of the fractional part each to the closest possible
/// representation.
///
/// If two representable values are equally close, the result of rounding is
/// the value with more trailing zeros in its significand bit pattern.
///
/// - Parameters:
/// - source: The rational value to convert to a floating-point value.
public init(_ source: Rational<Int16>) {
let (whole, fraction) = source.mixed
self = Self(whole) + Self(fraction.numerator) / Self(fraction.denominator)
}
/// Creates a new floating-point value from the given rational value, after
/// rounding the whole part, the numerator of the fractional part, and the
/// denominator of the fractional part each to the closest possible
/// representation.
///
/// If two representable values are equally close, the result of rounding is
/// the value with more trailing zeros in its significand bit pattern.
///
/// - Parameters:
/// - source: The rational value to convert to a floating-point value.
public init(_ source: Rational<Int32>) {
let (whole, fraction) = source.mixed
self = Self(whole) + Self(fraction.numerator) / Self(fraction.denominator)
}
/// Creates a new floating-point value from the given rational value, after
/// rounding the whole part, the numerator of the fractional part, and the
/// denominator of the fractional part each to the closest possible
/// representation.
///
/// If two representable values are equally close, the result of rounding is
/// the value with more trailing zeros in its significand bit pattern.
///
/// - Parameters:
/// - source: The rational value to convert to a floating-point value.
public init(_ source: Rational<Int64>) {
let (whole, fraction) = source.mixed
self = Self(whole) + Self(fraction.numerator) / Self(fraction.denominator)
}
}