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
|
#
#
# Nim's Runtime Library
# (c) Copyright 2015 Dennis Felsing
#
# See the file "copying.txt", included in this
# distribution, for details about the copyright.
#
## This module implements rational numbers, consisting of a numerator and
## a denominator. The denominator can not be 0.
runnableExamples:
let
r1 = 1 // 2
r2 = -3 // 4
doAssert r1 + r2 == -1 // 4
doAssert r1 - r2 == 5 // 4
doAssert r1 * r2 == -3 // 8
doAssert r1 / r2 == -2 // 3
import math, hashes
type Rational*[T] = object
## A rational number, consisting of a numerator `num` and a denominator `den`.
num*, den*: T
func reduce*[T: SomeInteger](x: var Rational[T]) =
## Reduces the rational number `x`, so that the numerator and denominator
## have no common divisors other than 1 (and -1).
## If `x` is 0, raises `DivByZeroDefect`.
##
## **Note:** This is called automatically by the various operations on rationals.
runnableExamples:
var r = Rational[int](num: 2, den: 4) # 1/2
reduce(r)
doAssert r.num == 1
doAssert r.den == 2
let common = gcd(x.num, x.den)
if x.den > 0:
x.num = x.num div common
x.den = x.den div common
elif x.den < 0:
x.num = -x.num div common
x.den = -x.den div common
else:
raise newException(DivByZeroDefect, "division by zero")
func initRational*[T: SomeInteger](num, den: T): Rational[T] =
## Creates a new rational number with numerator `num` and denominator `den`.
## `den` must not be 0.
##
## **Note:** `den != 0` is not checked when assertions are turned off.
assert(den != 0, "a denominator of zero is invalid")
result.num = num
result.den = den
reduce(result)
func `//`*[T](num, den: T): Rational[T] =
## A friendlier version of `initRational <#initRational,T,T>`_.
runnableExamples:
let x = 1 // 3 + 1 // 5
doAssert x == 8 // 15
initRational[T](num, den)
func `$`*[T](x: Rational[T]): string =
## Turns a rational number into a string.
runnableExamples:
doAssert $(1 // 2) == "1/2"
result = $x.num & "/" & $x.den
func toRational*[T: SomeInteger](x: T): Rational[T] =
## Converts some integer `x` to a rational number.
runnableExamples:
doAssert toRational(42) == 42 // 1
result.num = x
result.den = 1
func toRational*(x: float,
n: int = high(int) shr (sizeof(int) div 2 * 8)): Rational[int] =
## Calculates the best rational approximation of `x`,
## where the denominator is smaller than `n`
## (default is the largest possible `int` for maximal resolution).
##
## The algorithm is based on the theory of continued fractions.
# David Eppstein / UC Irvine / 8 Aug 1993
# With corrections from Arno Formella, May 2008
runnableExamples:
let x = 1.2
doAssert x.toRational.toFloat == x
var
m11, m22 = 1
m12, m21 = 0
ai = int(x)
x = x
while m21 * ai + m22 <= n:
swap m12, m11
swap m22, m21
m11 = m12 * ai + m11
m21 = m22 * ai + m21
if x == float(ai): break # division by zero
x = 1 / (x - float(ai))
if x > float(high(int32)): break # representation failure
ai = int(x)
result = m11 // m21
func toFloat*[T](x: Rational[T]): float =
## Converts a rational number `x` to a `float`.
x.num / x.den
func toInt*[T](x: Rational[T]): int =
## Converts a rational number `x` to an `int`. Conversion rounds towards 0 if
## `x` does not contain an integer value.
x.num div x.den
func `+`*[T](x, y: Rational[T]): Rational[T] =
## Adds two rational numbers.
let common = lcm(x.den, y.den)
result.num = common div x.den * x.num + common div y.den * y.num
result.den = common
reduce(result)
func `+`*[T](x: Rational[T], y: T): Rational[T] =
## Adds the rational `x` to the int `y`.
result.num = x.num + y * x.den
result.den = x.den
func `+`*[T](x: T, y: Rational[T]): Rational[T] =
## Adds the int `x` to the rational `y`.
result.num = x * y.den + y.num
result.den = y.den
func `+=`*[T](x: var Rational[T], y: Rational[T]) =
## Adds the rational `y` to the rational `x` in-place.
let common = lcm(x.den, y.den)
x.num = common div x.den * x.num + common div y.den * y.num
x.den = common
reduce(x)
func `+=`*[T](x: var Rational[T], y: T) =
## Adds the int `y` to the rational `x` in-place.
x.num += y * x.den
func `-`*[T](x: Rational[T]): Rational[T] =
## Unary minus for rational numbers.
result.num = -x.num
result.den = x.den
func `-`*[T](x, y: Rational[T]): Rational[T] =
## Subtracts two rational numbers.
let common = lcm(x.den, y.den)
result.num = common div x.den * x.num - common div y.den * y.num
result.den = common
reduce(result)
func `-`*[T](x: Rational[T], y: T): Rational[T] =
## Subtracts the int `y` from the rational `x`.
result.num = x.num - y * x.den
result.den = x.den
func `-`*[T](x: T, y: Rational[T]): Rational[T] =
## Subtracts the rational `y` from the int `x`.
result.num = x * y.den - y.num
result.den = y.den
func `-=`*[T](x: var Rational[T], y: Rational[T]) =
## Subtracts the rational `y` from the rational `x` in-place.
let common = lcm(x.den, y.den)
x.num = common div x.den * x.num - common div y.den * y.num
x.den = common
reduce(x)
func `-=`*[T](x: var Rational[T], y: T) =
## Subtracts the int `y` from the rational `x` in-place.
x.num -= y * x.den
func `*`*[T](x, y: Rational[T]): Rational[T] =
## Multiplies two rational numbers.
result.num = x.num * y.num
result.den = x.den * y.den
reduce(result)
func `*`*[T](x: Rational[T], y: T): Rational[T] =
## Multiplies the rational `x` with the int `y`.
result.num = x.num * y
result.den = x.den
reduce(result)
func `*`*[T](x: T, y: Rational[T]): Rational[T] =
## Multiplies the int `x` with the rational `y`.
result.num = x * y.num
result.den = y.den
reduce(result)
func `*=`*[T](x: var Rational[T], y: Rational[T]) =
## Multiplies the rational `x` by `y` in-place.
x.num *= y.num
x.den *= y.den
reduce(x)
func `*=`*[T](x: var Rational[T], y: T) =
## Multiplies the rational `x` by the int `y` in-place.
x.num *= y
reduce(x)
func reciprocal*[T](x: Rational[T]): Rational[T] =
## Calculates the reciprocal of `x` (`1/x`).
## If `x` is 0, raises `DivByZeroDefect`.
if x.num > 0:
result.num = x.den
result.den = x.num
elif x.num < 0:
result.num = -x.den
result.den = -x.num
else:
raise newException(DivByZeroDefect, "division by zero")
func `/`*[T](x, y: Rational[T]): Rational[T] =
## Divides the rational `x` by the rational `y`.
result.num = x.num * y.den
result.den = x.den * y.num
reduce(result)
func `/`*[T](x: Rational[T], y: T): Rational[T] =
## Divides the rational `x` by the int `y`.
result.num = x.num
result.den = x.den * y
reduce(result)
func `/`*[T](x: T, y: Rational[T]): Rational[T] =
## Divides the int `x` by the rational `y`.
result.num = x * y.den
result.den = y.num
reduce(result)
func `/=`*[T](x: var Rational[T], y: Rational[T]) =
## Divides the rational `x` by the rational `y` in-place.
x.num *= y.den
x.den *= y.num
reduce(x)
func `/=`*[T](x: var Rational[T], y: T) =
## Divides the rational `x` by the int `y` in-place.
x.den *= y
reduce(x)
func cmp*(x, y: Rational): int =
## Compares two rationals. Returns
## * a value less than zero, if `x < y`
## * a value greater than zero, if `x > y`
## * zero, if `x == y`
(x - y).num
func `<`*(x, y: Rational): bool =
## Returns true if `x` is less than `y`.
(x - y).num < 0
func `<=`*(x, y: Rational): bool =
## Returns tue if `x` is less than or equal to `y`.
(x - y).num <= 0
func `==`*(x, y: Rational): bool =
## Compares two rationals for equality.
(x - y).num == 0
func abs*[T](x: Rational[T]): Rational[T] =
## Returns the absolute value of `x`.
runnableExamples:
doAssert abs(1 // 2) == 1 // 2
doAssert abs(-1 // 2) == 1 // 2
result.num = abs x.num
result.den = abs x.den
func `div`*[T: SomeInteger](x, y: Rational[T]): T =
## Computes the rational truncated division.
(x.num * y.den) div (y.num * x.den)
func `mod`*[T: SomeInteger](x, y: Rational[T]): Rational[T] =
## Computes the rational modulo by truncated division (remainder).
## This is same as `x - (x div y) * y`.
result = ((x.num * y.den) mod (y.num * x.den)) // (x.den * y.den)
reduce(result)
func floorDiv*[T: SomeInteger](x, y: Rational[T]): T =
## Computes the rational floor division.
##
## Floor division is conceptually defined as `floor(x / y)`.
## This is different from the `div` operator, which is defined
## as `trunc(x / y)`. That is, `div` rounds towards 0 and `floorDiv`
## rounds down.
floorDiv(x.num * y.den, y.num * x.den)
func floorMod*[T: SomeInteger](x, y: Rational[T]): Rational[T] =
## Computes the rational modulo by floor division (modulo).
##
## This is same as `x - floorDiv(x, y) * y`.
## This func behaves the same as the `%` operator in Python.
result = floorMod(x.num * y.den, y.num * x.den) // (x.den * y.den)
reduce(result)
func hash*[T](x: Rational[T]): Hash =
## Computes the hash for the rational `x`.
# reduce first so that hash(x) == hash(y) for x == y
var copy = x
reduce(copy)
var h: Hash = 0
h = h !& hash(copy.num)
h = h !& hash(copy.den)
result = !$h
|