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Rational Real World Tests

The first set of tests measure the times taken to calculate the n'th Bernoulli number via mixed rational/integer arithmetic to give an exact rational result.

Table 1.18. Relative times taken to calculate Bm

m

cpp_rational (time in seconds)

mpq_rational (time relative to cpp_rational)

mpq_class (time relative to cpp_rational)

50

1.888453

1.7317576874

1.893438174

54

2.250503

2.1519411438

2.4936252029

58

2.734527

2.2056805437

1.9866752093

62

3.318122

2.5182796172

2.2662876772

66

3.887281

2.4263077457

2.588347485

70

4.628535

1.9756346231

2.5367158291

74

5.3541

1.7052929531

2.3345854579

78

6.321172

1.6669601776

2.0390342171

82

7.13052

1.7689101216

1.9837833706

86

8.390095

1.6245518078

1.5785492298

90

10.62176

1.6294440846

1.5779053566

94

11.364409

1.5786845581

1.5614997665

98

14.030636

1.2616490799

1.4799734666

102

15.268211

1.6145657798

1.6128342738

106

15.253028

1.8985381788

1.8206293859

110

17.637756

1.8283953469

1.6559522084

114

18.335007

1.8018346543

1.8343225612

118

21.044146

2.0462015897

1.8106934346

122

23.71295

2.0041556196

1.7127701108

126

25.993901

1.9901613075

1.6974264848

130

30.17278

1.8897211328

1.6405006433

134

43.992333

1.2223514038

1.2232961594

138

40.702777

1.3551305848

1.492072224

142

47.01495

1.4361410785

1.4005683724

146

51.468592

1.4279172043

1.4072223891

150

70.736106

1.3628087048

1.2309294917

154

74.638691

1.2509392749

1.2846060497

158

76.642396

1.3383691449

1.2555573941

162

104.906795

1.1722665057

1.0121124375

166

108.175914

0.9272805682

1.2472813403

170

125.363885

0.8114005082

1.1349673712

174

119.813754

0.9747547014

1.5677900135

178

130.672631

0.904429444

1.6089208382

182

136.002124

0.8359445107

1.3173987636

186

152.169271

0.7919228515

1.3523570012

190

149.444035

0.8353259734

1.4571606823

194

149.609183

0.8898088896

1.6132629038

198

167.594528

0.8947434429

1.326461858


In this use case, most the of the rational numbers are fairly small and so the times taken are dominated by the number of allocations performed. The following table illustrates how well each type performs on suppressing allocations:

Table 1.19. Total Allocation Counts for Bernoulli Number Calculation

m

cpp_rational

mpq_rational

mpq_class

2

0

77

101

4

0

187

252

6

0

345

471

8

0

551

758

10

0

805

1113

12

0

1107

1536

14

0

1457

2027

16

0

1857

2587

18

0

2336

3216

20

0

2885

3913

22

6

3511

4706

24

22

4203

5600

26

83

4963

6575

28

377

5806

7632

30

780

6738

8769

32

1454

7771

9988

34

2001

9357

11289

36

2789

10598

12704

38

3669

11948

14252

40

4653

13403

15891

42

5923

14976

17620

44

7379

16622

19449

46

8839

18367

21367

48

10296

20227

23431

50

12045

22857

25646

52

13603

25044

27962

54

15276

27331

30389

56

17239

29749

32919

58

19337

32257

35552

60

21409

34958

38417

62

23694

37800

41396

64

27923

39556

44498

66

30240

44706

47711

68

32566

47934

51042

70

35019

51417

54637

72

37460

55047

58363

74

40282

58777

62211

76

42914

62691

66183

78

45752

66694

70296

80

48681

70905

74620

82

51986

77633

79160

84

54855

82364

83842

86

59032

87239

88659

88

63595

92256

93618

90

68352

97486

98820

92

72446

102974

104256

94

76468

108620

109844

96

80361

111109

115594

98

84783

121460

121486

100

89044

127730

127633

102

93561

134241

134050

104

98452

140919

140616

106

103530

147763

147364

108

108326

154804

154276

110

113891

162184

161562

112

119213

169736

169038

114

124770

182417

176693

116

130624

190589

184519

118

137941

198975

192525

120

144829

207759

200952

122

152045

216736

209561

124

158610

225905

218371

126

165383

235283

227360

128

173971

230678

236670

130

181696

248593

246308

132

189310

258615

256148

134

197708

268800

266192

136

205800

279215

276436

138

212940

290112

287167

140

217502

301235

298101

142

223486

312564

309243

144

229579

324133

320605

146

237213

344671

332333

148

248799

357200

344420

150

261345

369947

356745

152

272741

382909

369279

154

283982

396162

382048

156

293626

409993

395353

158

304036

424022

408907

160

313869

427747

422690

162

323626

454723

436715

164

333294

469863

451304

166

343072

485238

466149

168

352236

500922

481221

170

362793

516840

496561

172

372645

533169

512315

174

382908

549962

528510

176

392507

567018

544947

178

404163

597694

561666

180

417539

615615

578647

182

432009

634079

596234

184

444684

652902

614114

186

457104

672042

632248

188

469331

691451

650654

190

481583

711512

669753

192

477225

698188

689111

194

488955

736905

708760

196

499797

757502

728675

198

511163

778591

749102


The second example measures the time taken to calculate the determinant of a 3x3 matrix of rational numbers. These numbers are randomly generated with n bits in both numerator and denominator. In this case the rate limiting step is the cost of calculating the GCD's during the computation:

Table 1.20. Relative Time Taken to Calculate 100 Determinants of Random 3x3 Matrixes

Bits

cpp_rational (ms)

mpq_rational (relative to cpp_rational)

mpq_class (relative to cpp_rational)

512

45

0.3111111111

0.3177777778

1024

103

0.3038834951

0.3038834951

2048

251

0.3087649402

0.3011952191

4096

667

0.2983508246

0.2893553223

8192

2033

0.261682243

0.2956222332

16384

6423

0.2358710883

0.3059318076

32768

24223

0.1875903067

0.1955992239


Table 1.21. Platform Details

Version

Compiler

GNU C++ version 10.3.0

GMP

6.2.0

MPFR

262146

Boost

107800

Run date

Sep 30 2021



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