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