cpython/Lib/fractions.py at main · python/cpython · GitHub

Repository navigation

History

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

864

865

866

867

868

869

870

871

872

873

874

875

876

877

878

879

880

881

882

883

884

885

886

887

888

889

890

891

892

893

894

895

896

897

898

899

900

901

902

903

904

905

906

907

908

909

910

911

912

913

914

915

916

917

918

919

920

921

922

923

924

925

926

927

928

929

930

931

932

933

934

935

936

937

938

939

940

941

942

943

944

945

946

947

948

949

950

951

952

953

954

955

956

957

958

959

960

961

962

963

964

965

966

967

968

969

970

971

972

973

974

975

976

977

978

979

980

981

982

983

984

985

986

987

988

989

990

991

992

993

994

995

996

997

998

999

1000

# Originally contributed by Sjoerd Mullender.

# Significantly modified by Jeffrey Yasskin <jyasskin at gmail.com>.

"""Fraction, infinite-precision, rational numbers."""

import functools

import math

import numbers

import operator

import re

import sys

__all__ = ['Fraction']

# Constants related to the hash implementation; hash(x) is based

# on the reduction of x modulo the prime _PyHASH_MODULUS.

_PyHASH_MODULUS = sys.hash_info.modulus

# Value to be used for rationals that reduce to infinity modulo

# _PyHASH_MODULUS.

_PyHASH_INF = sys.hash_info.inf

@functools.lru_cache(maxsize = 1 << 14)

def _hash_algorithm(numerator, denominator):

# To make sure that the hash of a Fraction agrees with the hash

# of a numerically equal integer, float or Decimal instance, we

# follow the rules for numeric hashes outlined in the

# documentation. (See library docs, 'Built-in Types').

try:

dinv = pow(denominator, -1, _PyHASH_MODULUS)

except ValueError:

# ValueError means there is no modular inverse.

hash_ = _PyHASH_INF

else:

# The general algorithm now specifies that the absolute value of

# the hash is

# (|N| * dinv) % P

# where N is self._numerator and P is _PyHASH_MODULUS. That's

# optimized here in two ways: first, for a non-negative int i,

# hash(i) == i % P, but the int hash implementation doesn't need

# to divide, and is faster than doing % P explicitly. So we do

# hash(|N| * dinv)

# instead. Second, N is unbounded, so its product with dinv may

# be arbitrarily expensive to compute. The final answer is the

# same if we use the bounded |N| % P instead, which can again

# be done with an int hash() call. If 0 <= i < P, hash(i) == i,

# so this nested hash() call wastes a bit of time making a

# redundant copy when |N| < P, but can save an arbitrarily large

# amount of computation for large |N|.

hash_ = hash(hash(abs(numerator)) * dinv)

result = hash_ if numerator >= 0 else -hash_

return -2 if result == -1 else result

_RATIONAL_FORMAT = re.compile(r"""

\A\s* # optional whitespace at the start,

(?P<sign>[-+]?) # an optional sign, then

(?=\d|\.\d) # lookahead for digit or .digit

(?P<num>\d*|\d+(_\d+)*) # numerator (possibly empty)

(?: # followed by

(?:\s*/\s*(?P<denom>\d+(_\d+)*))? # an optional denominator

| # or

(?:\.(?P<decimal>\d*|\d+(_\d+)*))? # an optional fractional part

(?:E(?P<exp>[-+]?\d+(_\d+)*))? # and optional exponent

)

\s*\z # and optional whitespace to finish

""", re.VERBOSE | re.IGNORECASE)

# Helpers for formatting

def _round_to_exponent(n, d, exponent, no_neg_zero=False):

"""Round a rational number to the nearest multiple of a given power of 10.

Rounds the rational number n/d to the nearest integer multiple of

10**exponent, rounding to the nearest even integer multiple in the case of

a tie. Returns a pair (sign: bool, significand: int) representing the

rounded value (-1)**sign * significand * 10**exponent.

If no_neg_zero is true, then the returned sign will always be False when

the significand is zero. Otherwise, the sign reflects the sign of the

input.

d must be positive, but n and d need not be relatively prime.

"""

if exponent >= 0:

d *= 10**exponent

else:

n *= 10**-exponent

# The divmod quotient is correct for round-ties-towards-positive-infinity;

# In the case of a tie, we zero out the least significant bit of q.

q, r = divmod(n + (d >> 1), d)

if r == 0 and d & 1 == 0:

q &= -2

sign = q < 0 if no_neg_zero else n < 0

return sign, abs(q)

def _round_to_figures(n, d, figures):

"""Round a rational number to a given number of significant figures.

Rounds the rational number n/d to the given number of significant figures

using the round-ties-to-even rule, and returns a triple

(sign: bool, significand: int, exponent: int) representing the rounded

value (-1)**sign * significand * 10**exponent.

In the special case where n = 0, returns a significand of zero and

an exponent of 1 - figures, for compatibility with formatting.

Otherwise, the returned significand satisfies

10**(figures - 1) <= significand < 10**figures.

d must be positive, but n and d need not be relatively prime.

figures must be positive.

"""

# Special case for n == 0.

if n == 0:

return False, 0, 1 - figures

# Find integer m satisfying 10**(m - 1) <= abs(n)/d <= 10**m. (If abs(n)/d

# is a power of 10, either of the two possible values for m is fine.)

str_n, str_d = str(abs(n)), str(d)

m = len(str_n) - len(str_d) + (str_d <= str_n)

# Round to a multiple of 10**(m - figures). The significand we get

# satisfies 10**(figures - 1) <= significand <= 10**figures.

exponent = m - figures

sign, significand = _round_to_exponent(n, d, exponent)

# Adjust in the case where significand == 10**figures, to ensure that

# 10**(figures - 1) <= significand < 10**figures.

if len(str(significand)) == figures + 1:

significand //= 10

exponent += 1

return sign, significand, exponent

# Pattern for matching non-float-style format specifications.

_GENERAL_FORMAT_SPECIFICATION_MATCHER = re.compile(r"""

(?:

(?P<fill>.)?

(?P<align>[<>=^])

(?P<sign>[-+ ]?)

# Alt flag forces a slash and denominator in the output, even for

# integer-valued Fraction objects.

(?P<alt>\#)?

# We don't implement the zeropad flag since there's no single obvious way

# to interpret it.

(?P<minimumwidth>0|[1-9][0-9]*)?

(?P<thousands_sep>[,_])?

""", re.DOTALL | re.VERBOSE).fullmatch

# Pattern for matching float-style format specifications;

# supports 'e', 'E', 'f', 'F', 'g', 'G' and '%' presentation types.

_FLOAT_FORMAT_SPECIFICATION_MATCHER = re.compile(r"""

(?:

(?P<fill>.)?

(?P<align>[<>=^])

(?P<sign>[-+ ]?)

(?P<no_neg_zero>z)?

(?P<alt>\#)?

# A '0' that's *not* followed by another digit is parsed as a minimum width

# rather than a zeropad flag.

(?P<zeropad>0(?=[0-9]))?

(?P<minimumwidth>[0-9]+)?

(?P<thousands_sep>[,_])?

(?:\.

(?=[,_0-9]) # lookahead for digit or separator

(?P<precision>[0-9]+)?

(?P<frac_separators>[,_])?

(?P<presentation_type>[eEfFgG%])

""", re.DOTALL | re.VERBOSE).fullmatch

class Fraction(numbers.Rational):

"""This class implements rational numbers.

In the two-argument form of the constructor, Fraction(8, 6) will

produce a rational number equivalent to 4/3. Both arguments must

be Rational. The numerator defaults to 0 and the denominator

defaults to 1 so that Fraction(3) == 3 and Fraction() == 0.

Fractions can also be constructed from:

- numeric strings similar to those accepted by the

float constructor (for example, '-2.3' or '1e10')

- strings of the form '123/456'

- float and Decimal instances

- other Rational instances (including integers)

"""

__slots__ = ('_numerator', '_denominator')

# We're immutable, so use __new__ not __init__

def __new__(cls, numerator=0, denominator=None):

"""Constructs a Rational.

Takes a string like '3/2' or '1.5', another Rational instance, a

numerator/denominator pair, or a float.

Examples

--------

>>> Fraction(10, -8)

Fraction(-5, 4)

>>> Fraction(Fraction(1, 7), 5)

Fraction(1, 35)

>>> Fraction(Fraction(1, 7), Fraction(2, 3))

Fraction(3, 14)

>>> Fraction('314')

Fraction(314, 1)

>>> Fraction('-35/4')

Fraction(-35, 4)

>>> Fraction('3.1415') # conversion from numeric string

Fraction(6283, 2000)

>>> Fraction('-47e-2') # string may include a decimal exponent

Fraction(-47, 100)

>>> Fraction(1.47) # direct construction from float (exact conversion)

Fraction(6620291452234629, 4503599627370496)

>>> Fraction(2.25)

Fraction(9, 4)

>>> Fraction(Decimal('1.47'))

Fraction(147, 100)

"""

self = super(Fraction, cls).__new__(cls)

if denominator is None:

if type(numerator) is int:

self._numerator = numerator

self._denominator = 1

return self

elif (isinstance(numerator, float) or

(not isinstance(numerator, type) and

hasattr(numerator, 'as_integer_ratio'))):

# Exact conversion

self._numerator, self._denominator = numerator.as_integer_ratio()

return self

elif isinstance(numerator, str):

# Handle construction from strings.

m = _RATIONAL_FORMAT.match(numerator)

if m is None:

raise ValueError('Invalid literal for Fraction: %r' %

numerator)

numerator = int(m.group('num') or '0')

denom = m.group('denom')

if denom:

denominator = int(denom)

else:

denominator = 1

decimal = m.group('decimal')

if decimal:

decimal = decimal.replace('_', '')

scale = 10**len(decimal)

numerator = numerator * scale + int(decimal)

denominator *= scale

exp = m.group('exp')

if exp:

exp = int(exp)

if exp >= 0:

numerator *= 10**exp

else:

denominator *= 10**-exp

if m.group('sign') == '-':

numerator = -numerator

elif isinstance(numerator, numbers.Rational):

self._numerator = numerator.numerator

self._denominator = numerator.denominator

return self

else:

raise TypeError("argument should be a string or a Rational "

"instance or have the as_integer_ratio() method")

elif type(numerator) is int is type(denominator):

pass # *very* normal case

elif (isinstance(numerator, numbers.Rational) and

isinstance(denominator, numbers.Rational)):

numerator, denominator = (

numerator.numerator * denominator.denominator,

denominator.numerator * numerator.denominator

)

else:

raise TypeError("both arguments should be "

"Rational instances")

if denominator == 0:

raise ZeroDivisionError('Fraction(%s, 0)' % numerator)

g = math.gcd(numerator, denominator)

if denominator < 0:

g = -g

numerator //= g

denominator //= g

self._numerator = numerator

self._denominator = denominator

return self

@classmethod

def from_number(cls, number):

"""Converts a finite real number to a rational number, exactly.

Beware that Fraction.from_number(0.3) != Fraction(3, 10).

"""

if type(number) is int:

return cls._from_coprime_ints(number, 1)

elif isinstance(number, numbers.Rational):

return cls._from_coprime_ints(number.numerator, number.denominator)

elif (isinstance(number, float) or

(not isinstance(number, type) and

hasattr(number, 'as_integer_ratio'))):

return cls._from_coprime_ints(*number.as_integer_ratio())

else:

raise TypeError("argument should be a Rational instance or "

"have the as_integer_ratio() method")

@classmethod

def from_float(cls, f):

"""Converts a finite float to a rational number, exactly.

Beware that Fraction.from_float(0.3) != Fraction(3, 10).

"""

if isinstance(f, float):

return cls._from_coprime_ints(*f.as_integer_ratio())

if isinstance(f, numbers.Integral):

return cls(f)

raise TypeError("%s.from_float() only takes floats, not %r (%s)" %

(cls.__name__, f, type(f).__name__))

@classmethod

def from_decimal(cls, dec):

"""Converts a finite Decimal instance to a rational number, exactly."""

from decimal import Decimal

if isinstance(dec, Decimal):

return cls._from_coprime_ints(*dec.as_integer_ratio())

if isinstance(dec, numbers.Integral):

dec = int(dec)

return cls._from_coprime_ints(*dec.as_integer_ratio())

raise TypeError("%s.from_decimal() only takes Decimals, not %r (%s)" %

(cls.__name__, dec, type(dec).__name__))

@classmethod

def _from_coprime_ints(cls, numerator, denominator, /):

"""Convert a pair of ints to a rational number, for internal use.

The ratio of integers should be in lowest terms and the denominator

should be positive.

"""

obj = super(Fraction, cls).__new__(cls)

obj._numerator = numerator

obj._denominator = denominator

return obj

def is_integer(self):

"""Return True if the Fraction is an integer."""

return self._denominator == 1

def as_integer_ratio(self):

"""Return a pair of integers, whose ratio is equal to the original Fraction.

The ratio is in lowest terms and has a positive denominator.

"""

return (self._numerator, self._denominator)

def limit_denominator(self, max_denominator=1000000):

"""Closest Fraction to self with denominator at most max_denominator.

>>> Fraction('3.141592653589793').limit_denominator(10)

Fraction(22, 7)

>>> Fraction('3.141592653589793').limit_denominator(100)

Fraction(311, 99)

>>> Fraction(4321, 8765).limit_denominator(10000)

Fraction(4321, 8765)

"""

# Algorithm notes: For any real number x, define a *best upper

# approximation* to x to be a rational number p/q such that:

# (1) p/q >= x, and

# (2) if p/q > r/s >= x then s > q, for any rational r/s.

# Define *best lower approximation* similarly. Then it can be

# proved that a rational number is a best upper or lower

# approximation to x if, and only if, it is a convergent or

# semiconvergent of the (unique shortest) continued fraction

# associated to x.

# To find a best rational approximation with denominator <= M,

# we find the best upper and lower approximations with

# denominator <= M and take whichever of these is closer to x.

# In the event of a tie, the bound with smaller denominator is

# chosen. If both denominators are equal (which can happen

# only when max_denominator == 1 and self is midway between

# two integers) the lower bound---i.e., the floor of self, is

# taken.

if max_denominator < 1:

raise ValueError("max_denominator should be at least 1")

if self._denominator <= max_denominator:

return Fraction(self)

p0, q0, p1, q1 = 0, 1, 1, 0

n, d = self._numerator, self._denominator

while True:

a = n//d

q2 = q0+a*q1

if q2 > max_denominator:

break

p0, q0, p1, q1 = p1, q1, p0+a*p1, q2

n, d = d, n-a*d

k = (max_denominator-q0)//q1

# Determine which of the candidates (p0+k*p1)/(q0+k*q1) and p1/q1 is

# closer to self. The distance between them is 1/(q1*(q0+k*q1)), while

# the distance from p1/q1 to self is d/(q1*self._denominator). So we

# need to compare 2*(q0+k*q1) with self._denominator/d.

if 2*d*(q0+k*q1) <= self._denominator:

return Fraction._from_coprime_ints(p1, q1)

else:

return Fraction._from_coprime_ints(p0+k*p1, q0+k*q1)

@property

def numerator(a):

return a._numerator

@property

def denominator(a):

return a._denominator

def __repr__(self):

"""repr(self)"""

return '%s(%s, %s)' % (self.__class__.__name__,

self._numerator, self._denominator)

def __str__(self):

"""str(self)"""

if self._denominator == 1:

return str(self._numerator)

else:

return '%s/%s' % (self._numerator, self._denominator)

def _format_general(self, match):

"""Helper method for __format__.

Handles fill, alignment, signs, and thousands separators in the

case of no presentation type.

"""

# Validate and parse the format specifier.

fill = match["fill"] or " "

align = match["align"] or ">"

pos_sign = "" if match["sign"] == "-" else match["sign"]

alternate_form = bool(match["alt"])

minimumwidth = int(match["minimumwidth"] or "0")

thousands_sep = match["thousands_sep"] or ''

# Determine the body and sign representation.

n, d = self._numerator, self._denominator

if d > 1 or alternate_form:

body = f"{abs(n):{thousands_sep}}/{d:{thousands_sep}}"

else:

body = f"{abs(n):{thousands_sep}}"

sign = '-' if n < 0 else pos_sign

# Pad with fill character if necessary and return.

padding = fill * (minimumwidth - len(sign) - len(body))

if align == ">":

return padding + sign + body

elif align == "<":

return sign + body + padding

elif align == "^":

half = len(padding) // 2

return padding[:half] + sign + body + padding[half:]

else: # align == "="

return sign + padding + body

def _format_float_style(self, match):

"""Helper method for __format__; handles float presentation types."""

fill = match["fill"] or " "

align = match["align"] or ">"

pos_sign = "" if match["sign"] == "-" else match["sign"]

no_neg_zero = bool(match["no_neg_zero"])

alternate_form = bool(match["alt"])

zeropad = bool(match["zeropad"])

minimumwidth = int(match["minimumwidth"] or "0")

thousands_sep = match["thousands_sep"]

precision = int(match["precision"] or "6")

frac_sep = match["frac_separators"] or ""

presentation_type = match["presentation_type"]

trim_zeros = presentation_type in "gG" and not alternate_form

trim_point = not alternate_form

exponent_indicator = "E" if presentation_type in "EFG" else "e"

if align == '=' and fill == '0':

zeropad = True

# Round to get the digits we need, figure out where to place the point,

# and decide whether to use scientific notation. 'point_pos' is the

# relative to the _end_ of the digit string: that is, it's the number

# of digits that should follow the point.

if presentation_type in "fF%":

exponent = -precision

if presentation_type == "%":

exponent -= 2

negative, significand = _round_to_exponent(

self._numerator, self._denominator, exponent, no_neg_zero)

scientific = False

point_pos = precision

else: # presentation_type in "eEgG"

figures = (

max(precision, 1)

if presentation_type in "gG"

else precision + 1

)

negative, significand, exponent = _round_to_figures(

self._numerator, self._denominator, figures)

scientific = (

presentation_type in "eE"

or exponent > 0

or exponent + figures <= -4

)

point_pos = figures - 1 if scientific else -exponent

# Get the suffix - the part following the digits, if any.

if presentation_type == "%":

suffix = "%"

elif scientific:

suffix = f"{exponent_indicator}{exponent + point_pos:+03d}"

else:

suffix = ""

# String of output digits, padded sufficiently with zeros on the left

# so that we'll have at least one digit before the decimal point.

digits = f"{significand:0{point_pos + 1}d}"

# Before padding, the output has the form f"{sign}{leading}{trailing}",

# where `leading` includes thousands separators if necessary and

# `trailing` includes the decimal separator where appropriate.

sign = "-" if negative else pos_sign

leading = digits[: len(digits) - point_pos]

frac_part = digits[len(digits) - point_pos :]

if trim_zeros:

frac_part = frac_part.rstrip("0")

separator = "" if trim_point and not frac_part else "."

if frac_sep:

frac_part = frac_sep.join(frac_part[pos:pos + 3]

for pos in range(0, len(frac_part), 3))

trailing = separator + frac_part + suffix

# Do zero padding if required.

if zeropad:

min_leading = minimumwidth - len(sign) - len(trailing)

# When adding thousands separators, they'll be added to the

# zero-padded portion too, so we need to compensate.

leading = leading.zfill(

3 * min_leading // 4 + 1 if thousands_sep else min_leading

)

# Insert thousands separators if required.

if thousands_sep:

first_pos = 1 + (len(leading) - 1) % 3

leading = leading[:first_pos] + "".join(

thousands_sep + leading[pos : pos + 3]

for pos in range(first_pos, len(leading), 3)

)

# We now have a sign and a body. Pad with fill character if necessary

# and return.

body = leading + trailing

padding = fill * (minimumwidth - len(sign) - len(body))

if align == ">":

return padding + sign + body

elif align == "<":

return sign + body + padding

elif align == "^":

half = len(padding) // 2

return padding[:half] + sign + body + padding[half:]

else: # align == "="

return sign + padding + body

def __format__(self, format_spec, /):

"""Format this fraction according to the given format specification."""

if match := _GENERAL_FORMAT_SPECIFICATION_MATCHER(format_spec):

return self._format_general(match)

if match := _FLOAT_FORMAT_SPECIFICATION_MATCHER(format_spec):

# Refuse the temptation to guess if both alignment _and_

# zero padding are specified.

if match["align"] is None or match["zeropad"] is None:

return self._format_float_style(match)

raise ValueError(

f"Invalid format specifier {format_spec!r} "

f"for object of type {type(self).__name__!r}"

)

def _operator_fallbacks(monomorphic_operator, fallback_operator,

handle_complex=True):

"""Generates forward and reverse operators given a purely-rational

operator and a function from the operator module.

Use this like:

__op__, __rop__ = _operator_fallbacks(just_rational_op, operator.op)

In general, we want to implement the arithmetic operations so

that mixed-mode operations either call an implementation whose

author knew about the types of both arguments, or convert both

to the nearest built in type and do the operation there. In

Fraction, that means that we define __add__ and __radd__ as:

def __add__(self, other):

# Both types have numerators/denominator attributes,

# so do the operation directly

if isinstance(other, (int, Fraction)):

return Fraction(self.numerator * other.denominator +

other.numerator * self.denominator,

self.denominator * other.denominator)

# float and complex don't have those operations, but we

# know about those types, so special case them.

elif isinstance(other, float):

return float(self) + other

elif isinstance(other, complex):

return complex(self) + other

# Let the other type take over.

return NotImplemented

def __radd__(self, other):

# radd handles more types than add because there's

# nothing left to fall back to.

if isinstance(other, numbers.Rational):

return Fraction(self.numerator * other.denominator +

other.numerator * self.denominator,

self.denominator * other.denominator)

elif isinstance(other, Real):

return float(other) + float(self)

elif isinstance(other, Complex):

return complex(other) + complex(self)

return NotImplemented

There are 5 different cases for a mixed-type addition on

Fraction. I'll refer to all of the above code that doesn't

refer to Fraction, float, or complex as "boilerplate". 'r'

will be an instance of Fraction, which is a subtype of

Rational (r : Fraction <: Rational), and b : B <:

Complex. The first three involve 'r + b':

1. If B <: Fraction, int, float, or complex, we handle

that specially, and all is well.

2. If Fraction falls back to the boilerplate code, and it

were to return a value from __add__, we'd miss the

possibility that B defines a more intelligent __radd__,

so the boilerplate should return NotImplemented from

__add__. In particular, we don't handle Rational

here, even though we could get an exact answer, in case

the other type wants to do something special.

3. If B <: Fraction, Python tries B.__radd__ before

Fraction.__add__. This is ok, because it was

implemented with knowledge of Fraction, so it can

handle those instances before delegating to Real or

Complex.

The next two situations describe 'b + r'. We assume that b

didn't know about Fraction in its implementation, and that it

uses similar boilerplate code:

4. If B <: Rational, then __radd_ converts both to the

builtin rational type (hey look, that's us) and

proceeds.

5. Otherwise, __radd__ tries to find the nearest common

base ABC, and fall back to its builtin type. Since this

class doesn't subclass a concrete type, there's no

implementation to fall back to, so we need to try as

hard as possible to return an actual value, or the user

will get a TypeError.

"""

def forward(a, b):

if isinstance(b, Fraction):

return monomorphic_operator(a, b)

elif isinstance(b, int):

return monomorphic_operator(a, Fraction(b))

elif isinstance(b, float):

return fallback_operator(float(a), b)

elif handle_complex and isinstance(b, complex):

return fallback_operator(float(a), b)

else:

return NotImplemented

forward.__name__ = '__' + fallback_operator.__name__ + '__'

forward.__doc__ = monomorphic_operator.__doc__

def reverse(b, a):

if isinstance(a, numbers.Rational):

# Includes ints.

return monomorphic_operator(Fraction(a), b)

elif isinstance(a, numbers.Real):

return fallback_operator(float(a), float(b))

elif handle_complex and isinstance(a, numbers.Complex):

return fallback_operator(complex(a), float(b))

else:

return NotImplemented

reverse.__name__ = '__r' + fallback_operator.__name__ + '__'

reverse.__doc__ = monomorphic_operator.__doc__

return forward, reverse

# Rational arithmetic algorithms: Knuth, TAOCP, Volume 2, 4.5.1.

# Assume input fractions a and b are normalized.

# 1) Consider addition/subtraction.

# Let g = gcd(da, db). Then

# na nb na*db ± nb*da

# a ± b == -- ± -- == ------------- ==

# da db da*db

# na*(db//g) ± nb*(da//g) t

# == ----------------------- == -

# (da*db)//g d

# Now, if g > 1, we're working with smaller integers.

# Note, that t, (da//g) and (db//g) are pairwise coprime.

# Indeed, (da//g) and (db//g) share no common factors (they were

# removed) and da is coprime with na (since input fractions are

# normalized), hence (da//g) and na are coprime. By symmetry,

# (db//g) and nb are coprime too. Then,

# gcd(t, da//g) == gcd(na*(db//g), da//g) == 1

# gcd(t, db//g) == gcd(nb*(da//g), db//g) == 1

# Above allows us optimize reduction of the result to lowest

# terms. Indeed,

# g2 = gcd(t, d) == gcd(t, (da//g)*(db//g)*g) == gcd(t, g)

# t//g2 t//g2

# a ± b == ----------------------- == ----------------

# (da//g)*(db//g)*(g//g2) (da//g)*(db//g2)

# is a normalized fraction. This is useful because the unnormalized

# denominator d could be much larger than g.

# We should special-case g == 1 (and g2 == 1), since 60.8% of

# randomly-chosen integers are coprime:

# https://en.wikipedia.org/wiki/Coprime_integers#Probability_of_coprimality

# Note, that g2 == 1 always for fractions, obtained from floats: here

# g is a power of 2 and the unnormalized numerator t is an odd integer.

# 2) Consider multiplication

# Let g1 = gcd(na, db) and g2 = gcd(nb, da), then

# na*nb na*nb (na//g1)*(nb//g2)

# a*b == ----- == ----- == -----------------

# da*db db*da (db//g1)*(da//g2)

# Note, that after divisions we're multiplying smaller integers.

# Also, the resulting fraction is normalized, because each of

# two factors in the numerator is coprime to each of the two factors

# in the denominator.

# Indeed, pick (na//g1). It's coprime with (da//g2), because input

# fractions are normalized. It's also coprime with (db//g1), because

# common factors are removed by g1 == gcd(na, db).

# As for addition/subtraction, we should special-case g1 == 1

# and g2 == 1 for same reason. That happens also for multiplying

# rationals, obtained from floats.

def _add(a, b):

"""a + b"""

na, da = a._numerator, a._denominator

nb, db = b._numerator, b._denominator

g = math.gcd(da, db)

if g == 1:

return Fraction._from_coprime_ints(na * db + da * nb, da * db)

s = da // g

t = na * (db // g) + nb * s

g2 = math.gcd(t, g)

if g2 == 1:

return Fraction._from_coprime_ints(t, s * db)

return Fraction._from_coprime_ints(t // g2, s * (db // g2))

__add__, __radd__ = _operator_fallbacks(_add, operator.add)

def _sub(a, b):

"""a - b"""

na, da = a._numerator, a._denominator

nb, db = b._numerator, b._denominator

g = math.gcd(da, db)

if g == 1:

return Fraction._from_coprime_ints(na * db - da * nb, da * db)

s = da // g

t = na * (db // g) - nb * s

g2 = math.gcd(t, g)

if g2 == 1:

return Fraction._from_coprime_ints(t, s * db)

return Fraction._from_coprime_ints(t // g2, s * (db // g2))

__sub__, __rsub__ = _operator_fallbacks(_sub, operator.sub)

def _mul(a, b):

"""a * b"""

na, da = a._numerator, a._denominator

nb, db = b._numerator, b._denominator

g1 = math.gcd(na, db)

if g1 > 1:

na //= g1

db //= g1

g2 = math.gcd(nb, da)

if g2 > 1:

nb //= g2

da //= g2

return Fraction._from_coprime_ints(na * nb, db * da)

__mul__, __rmul__ = _operator_fallbacks(_mul, operator.mul)

def _div(a, b):

"""a / b"""

# Same as _mul(), with inversed b.

nb, db = b._numerator, b._denominator

if nb == 0:

raise ZeroDivisionError('Fraction(%s, 0)' % db)

na, da = a._numerator, a._denominator

g1 = math.gcd(na, nb)

if g1 > 1:

na //= g1

nb //= g1

g2 = math.gcd(db, da)

if g2 > 1:

da //= g2

db //= g2

n, d = na * db, nb * da

if d < 0:

n, d = -n, -d

return Fraction._from_coprime_ints(n, d)

__truediv__, __rtruediv__ = _operator_fallbacks(_div, operator.truediv)

def _floordiv(a, b):

"""a // b"""

return (a.numerator * b.denominator) // (a.denominator * b.numerator)

__floordiv__, __rfloordiv__ = _operator_fallbacks(_floordiv, operator.floordiv, False)

def _divmod(a, b):

"""(a // b, a % b)"""

da, db = a.denominator, b.denominator

div, n_mod = divmod(a.numerator * db, da * b.numerator)

return div, Fraction(n_mod, da * db)

__divmod__, __rdivmod__ = _operator_fallbacks(_divmod, divmod, False)

def _mod(a, b):

"""a % b"""

da, db = a.denominator, b.denominator

return Fraction((a.numerator * db) % (b.numerator * da), da * db)

__mod__, __rmod__ = _operator_fallbacks(_mod, operator.mod, False)

def __pow__(a, b, modulo=None):

"""a ** b

If b is not an integer, the result will be a float or complex

since roots are generally irrational. If b is an integer, the

result will be rational.

"""

if modulo is not None:

return NotImplemented

if isinstance(b, numbers.Rational):

if b.denominator == 1:

power = b.numerator

if power >= 0:

return Fraction._from_coprime_ints(a._numerator ** power,

a._denominator ** power)

elif a._numerator > 0:

return Fraction._from_coprime_ints(a._denominator ** -power,

a._numerator ** -power)

elif a._numerator == 0:

raise ZeroDivisionError('Fraction(%s, 0)' %

a._denominator ** -power)

else:

return Fraction._from_coprime_ints((-a._denominator) ** -power,

(-a._numerator) ** -power)

else:

# A fractional power will generally produce an

# irrational number.

return float(a) ** float(b)

elif isinstance(b, (float, complex)):

return float(a) ** b

else:

return NotImplemented

def __rpow__(b, a, modulo=None):

"""a ** b"""

if modulo is not None:

return NotImplemented

if b._denominator == 1 and b._numerator >= 0:

# If a is an int, keep it that way if possible.

return a ** b._numerator

if isinstance(a, numbers.Rational):

return Fraction(a.numerator, a.denominator) ** b

if b._denominator == 1:

return a ** b._numerator

return a ** float(b)

def __pos__(a):

"""+a: Coerces a subclass instance to Fraction"""

return Fraction._from_coprime_ints(a._numerator, a._denominator)

def __neg__(a):

"""-a"""

return Fraction._from_coprime_ints(-a._numerator, a._denominator)

def __abs__(a):

"""abs(a)"""

return Fraction._from_coprime_ints(abs(a._numerator), a._denominator)

def __int__(a, _index=operator.index):

"""int(a)"""

if a._numerator < 0:

return _index(-(-a._numerator // a._denominator))

else:

return _index(a._numerator // a._denominator)

def __trunc__(a):

"""math.trunc(a)"""

if a._numerator < 0:

return -(-a._numerator // a._denominator)

else:

return a._numerator // a._denominator

def __floor__(a):

"""math.floor(a)"""

return a._numerator // a._denominator

def __ceil__(a):

"""math.ceil(a)"""

# The negations cleverly convince floordiv to return the ceiling.

return -(-a._numerator // a._denominator)

def __round__(self, ndigits=None):

"""round(self, ndigits)

Rounds half toward even.

"""

if ndigits is None:

d = self._denominator

floor, remainder = divmod(self._numerator, d)

if remainder * 2 < d:

return floor

elif remainder * 2 > d:

return floor + 1

# Deal with the half case:

elif floor % 2 == 0:

return floor

else:

return floor + 1

shift = 10**abs(ndigits)

# See _operator_fallbacks.forward to check that the results of

# these operations will always be Fraction and therefore have

# round().

if ndigits > 0:

return Fraction(round(self * shift), shift)

else:

return Fraction(round(self / shift) * shift)

def __hash__(self):

"""hash(self)"""

return _hash_algorithm(self._numerator, self._denominator)

def __eq__(a, b):

View remainder of file in raw view

cpython/Lib/fractions.py at main · python/cpython · GitHub

Repository navigation

Expand file tree

Breadcrumbs

fractions.py

Latest commit

History

Breadcrumbs

fractions.py

File metadata and controls