# Systematically avoiding arbitarily large exponents in calculations with small answers

• kmarinas86
In summary, avoiding arbitrarily large exponents can improve the accuracy and efficiency of calculations by preventing overflow errors and loss of precision. This can be achieved by breaking down calculations into smaller steps or using scientific notation. It is important to avoid arbitrarily large exponents in scientific calculations to ensure reliable and accurate results. Consistently avoiding arbitrarily large exponents can be achieved by setting clear guidelines and protocols for handling calculations with small answers.
kmarinas86
$$y=\frac{100}{100}\frac{\frac{(((100/100)^2-100*100)e^{x/(100*100)}+(100*100))e^{-x/(100*100)})-(100/100)^2e^{-x*100/100}}{100/100-100*100}}{1}$$

http://www.quickmath.com/msolver//graphs/2011-12-06/e5/61/e2/e561e26ac85a4b1da0176565a4827772-2.png?t=1323201139

Code:
0	0
1	3.67866E-05
2	0.000113525
3	0.000204954
4	0.000301782
5	0.000400589
6	0.000500118
7	0.000599906
8	0.000699784
9	0.000799687
10	0.000899595
11	0.000999497
12	0.001099391
13	0.001199276
14	0.00129915
15	0.001399015
16	0.001498871
17	0.001598716
18	0.001698551
19	0.001798376
20	0.001898191
21	0.001997996
22	0.002097792
23	0.002197577
24	0.002297352
25	0.002397117
26	0.002496873
27	0.002596618
28	0.002696353
29	0.002796079
30	0.002895794
31	0.0029955
32	0.003095195
33	0.00319488
34	0.003294556
35	0.003394222
36	0.003493877
37	0.003593523
38	0.003693158
39	0.003792784
40	0.0038924
41	0.003992006
42	0.004091601
43	0.004191187
44	0.004290763
45	0.004390329
46	0.004489885
47	0.004589431
48	0.004688967
49	0.004788493
50	0.00488801
51	0.004987516
52	0.005087012
53	0.005186498
54	0.005285975
55	0.005385441
56	0.005484898
57	0.005584344
58	0.005683781
59	0.005783208
60	0.005882624
61	0.005982031
62	0.006081428
63	0.006180815
64	0.006280192
65	0.006379559
66	0.006478916
67	0.006578263
68	0.0066776
69	0.006776927
70	0.006876245
71	0.006975552
72	0.00707485
73	0.007174137
74	0.007273415
75	0.007372682
76	0.00747194
77	0.007571188
78	0.007670426
79	0.007769654
80	0.007868872
81	0.00796808
82	0.008067278
83	0.008166467
84	0.008265645
85	0.008364814
86	0.008463972
87	0.008563121
88	0.00866226
89	0.008761388
90	0.008860507
91	0.008959616
92	0.009058715
93	0.009157805
94	0.009256884
95	0.009355953
96	0.009455013
97	0.009554062
98	0.009653102
99	0.009752132
100	0.009851151
101	0.009950161
102	0.010049161
103	0.010148151
104	0.010247132
105	0.010346102
106	0.010445062
107	0.010544013
108	0.010642954
109	0.010741884
110	0.010840805
111	0.010939716
112	0.011038617
113	0.011137509
114	0.01123639
115	0.011335261
116	0.011434123
117	0.011532974
118	0.011631816
119	0.011730648
120	0.01182947
121	0.011928282
122	0.012027084
123	0.012125877
124	0.012224659
125	0.012323432
126	0.012422195
127	0.012520947
128	0.01261969
129	0.012718423
130	0.012817147
131	0.01291586
132	0.013014564
133	0.013113257
134	0.013211941
135	0.013310615
136	0.013409279
137	0.013507933
138	0.013606577
139	0.013705212
140	0.013803836
141	0.013902451
142	0.014001056
143	0.014099651
144	0.014198236
145	0.014296811
146	0.014395376
147	0.014493932
148	0.014592478
149	0.014691013
150	0.014789539
151	0.014888055
152	0.014986562
153	0.015085058
154	0.015183545
155	0.015282021
156	0.015380488
157	0.015478945
158	0.015577393
159	0.01567583
160	0.015774257
161	0.015872675
162	0.015971083
163	0.016069481
164	0.016167869
165	0.016266247
166	0.016364616
167	0.016462974
168	0.016561323
169	0.016659662
170	0.016757991
171	0.01685631
172	0.01695462
173	0.01705292
174	0.017151209
175	0.017249489
176	0.017347759
177	0.01744602
178	0.01754427
179	0.017642511
180	0.017740742
181	0.017838963
182	0.017937174
183	0.018035375
184	0.018133567
185	0.018231749
186	0.01832992
187	0.018428083
188	0.018526235
189	0.018624377
190	0.01872251
191	0.018820633
192	0.018918746
193	0.019016849
194	0.019114943
195	0.019213026
196	0.0193111
197	0.019409164
198	0.019507218
199	0.019605262
200	0.019703297
201	0.019801322
202	0.019899337
203	0.019997342
204	0.020095337
205	0.020193323
206	0.020291299
207	0.020389265
208	0.020487221
209	0.020585167
210	0.020683104
211	0.020781031
212	0.020878948
213	0.020976855
214	0.021074752
215	0.02117264
216	0.021270518
217	0.021368386
218	0.021466244
219	0.021564092
220	0.021661931
221	0.02175976
222	0.021857579
223	0.021955389
224	0.022053188
225	0.022150978
226	0.022248758
227	0.022346528
228	0.022444289
229	0.022542039
230	0.02263978
231	0.022737511
232	0.022835233
233	0.022932944
234	0.023030646
235	0.023128338
236	0.02322602
237	0.023323693
238	0.023421356
239	0.023519009
240	0.023616652
241	0.023714285
242	0.023811909
243	0.023909523
244	0.024007127
245	0.024104722
246	0.024202306
247	0.024299881
248	0.024397446
249	0.024495002
250	0.024592547
251	0.024690083
252	0.024787609
253	0.024885126
254	0.024982632
255	0.025080129
256	0.025177616
257	0.025275094
258	0.025372561
259	0.025470019
260	0.025567467
261	0.025664906
262	0.025762334
263	0.025859753
264	0.025957162
265	0.026054562
266	0.026151951
267	0.026249331
268	0.026346701
269	0.026444062
270	0.026541413
271	0.026638754
272	0.026736085
273	0.026833406
274	0.026930718
275	0.02702802
276	0.027125313
277	0.027222595
278	0.027319868
279	0.027417131
280	0.027514385
281	0.027611628
282	0.027708862
283	0.027806087
284	0.027903301
285	0.028000506
286	0.028097701
287	0.028194886
288	0.028292062
289	0.028389228
290	0.028486384
291	0.028583531
292	0.028680667
293	0.028777795
294	0.028874912
295	0.02897202
296	0.029069118
297	0.029166206
298	0.029263284
299	0.029360353
300	0.029457412
301	0.029554462
302	0.029651501
303	0.029748531
304	0.029845552
305	0.029942562
306	0.030039563
307	0.030136554
308	0.030233536
309	0.030330508
310	0.03042747
311	0.030524422
312	0.030621365
313	0.030718298
314	0.030815221
315	0.030912135
316	0.031009039
317	0.031105933
318	0.031202818
319	0.031299692
320	0.031396558
321	0.031493413
322	0.031590259
323	0.031687095
324	0.031783921
325	0.031880738
326	0.031977545
327	0.032074343
328	0.03217113
329	0.032267909
330	0.032364677
331	0.032461436
332	0.032558185
333	0.032654924
334	0.032751654
335	0.032848374
336	0.032945084
337	0.033041785
338	0.033138476
339	0.033235157
340	0.033331829
341	0.033428491
342	0.033525143
343	0.033621786
344	0.033718418
345	0.033815042
346	0.033911655
347	0.034008259
348	0.034104854
349	0.034201439
350	0.034298014
351	0.034394579
352	0.034491135
353	0.034587681
354	0.034684217
355	0.034780744
356	0.034877261
357	0.034973768
358	0.035070266
359	0.035166754
360	0.035263233
361	0.035359702
362	0.035456161
363	0.03555261
364	0.03564905
365	0.035745481
366	0.035841901
367	0.035938312
368	0.036034714
369	0.036131105
370	0.036227487
371	0.03632386
372	0.036420223
373	0.036516576
374	0.036612919
375	0.036709253
376	0.036805577
377	0.036901892
378	0.036998197
379	0.037094492
380	0.037190778
381	0.037287054
382	0.037383321
383	0.037479578
384	0.037575825
385	0.037672062
386	0.03776829
387	0.037864509
388	0.037960718
389	0.038056917
390	0.038153106
391	0.038249286
392	0.038345456
393	0.038441617
394	0.038537768
395	0.038633909
396	0.038730041
397	0.038826163
398	0.038922276
399	0.039018379
400	0.039114472
401	0.039210556
402	0.03930663
403	0.039402695
404	0.03949875
405	0.039594795
406	0.039690831
407	0.039786857
408	0.039882873
409	0.03997888
410	0.040074878
411	0.040170865
412	0.040266843
413	0.040362812
414	0.040458771
415	0.04055472
416	0.04065066
417	0.04074659
418	0.040842511
419	0.040938421
420	0.041034323
421	0.041130215
422	0.041226097
423	0.041321969
424	0.041417832
425	0.041513686
426	0.04160953
427	0.041705364
428	0.041801189
429	0.041897004
430	0.041992809
431	0.042088605
432	0.042184391
433	0.042280168
434	0.042375935
435	0.042471693
436	0.042567441
437	0.04266318
438	0.042758908
439	0.042854628
440	0.042950338
441	0.043046038
442	0.043141728
443	0.043237409
444	0.043333081
445	0.043428743
446	0.043524395
447	0.043620038
448	0.043715671
449	0.043811295
450	0.043906909
451	0.044002513
452	0.044098108
453	0.044193694
454	0.04428927
455	0.044384836
456	0.044480393
457	0.04457594
458	0.044671477
459	0.044767006
460	0.044862524
461	0.044958033
462	0.045053532
463	0.045149022
464	0.045244503
465	0.045339973
466	0.045435435
467	0.045530886
468	0.045626328
469	0.045721761
470	0.045817184
471	0.045912598
472	0.046008002
473	0.046103396
474	0.046198781
475	0.046294156
476	0.046389522
477	0.046484878
478	0.046580225
479	0.046675562
480	0.04677089
481	0.046866208
482	0.046961517
483	0.047056816
484	0.047152105
485	0.047247385
486	0.047342656
487	0.047437917
488	0.047533168
489	0.04762841
490	0.047723643
491	0.047818866
492	0.047914079
493	0.048009283
494	0.048104477
495	0.048199662
496	0.048294837
497	0.048390003
498	0.048485159
499	0.048580306
500	0.048675443
501	0.048770571
502	0.048865689
503	0.048960798
504	0.049055897
505	0.049150986
506	0.049246067
507	0.049341137
508	0.049436198
509	0.04953125
510	0.049626292
511	0.049721325
512	0.049816348
513	0.049911361
514	0.050006366
515	0.05010136
516	0.050196345
517	0.050291321
518	0.050386287
519	0.050481244
520	0.050576191
521	0.050671128
522	0.050766057
523	0.050860975
524	0.050955884
525	0.051050784
526	0.051145674
527	0.051240555
528	0.051335426
529	0.051430288
530	0.05152514
531	0.051619983
532	0.051714816
533	0.05180964
534	0.051904454
535	0.051999259
536	0.052094054
537	0.05218884
538	0.052283616
539	0.052378383
540	0.052473141
541	0.052567889
542	0.052662627
543	0.052757356
544	0.052852076
545	0.052946786
546	0.053041486
547	0.053136178
548	0.053230859
549	0.053325531
550	0.053420194
551	0.053514847
552	0.053609491
553	0.053704125
554	0.05379875
555	0.053893366
556	0.053987972
557	0.054082568
558	0.054177155
559	0.054271733
560	0.054366301
561	0.054460859
562	0.054555409
563	0.054649948
564	0.054744479
565	0.054838999
566	0.054933511
567	0.055028013
568	0.055122505
569	0.055216988
570	0.055311462
571	0.055405926
572	0.055500381
573	0.055594826
574	0.055689262
575	0.055783688
576	0.055878105
577	0.055972512
578	0.05606691
579	0.056161299
580	0.056255678
581	0.056350048
582	0.056444408
583	0.056538759
584	0.0566331
585	0.056727432
586	0.056821755
587	0.056916068
588	0.057010372
589	0.057104666
590	0.057198951
591	0.057293226
592	0.057387492
593	0.057481749
594	0.057575996
595	0.057670233
596	0.057764462
597	0.057858681
598	0.05795289
599	0.05804709
600	0.058141281
601	0.058235462
602	0.058329633
603	0.058423796
604	0.058517949
605	0.058612092
606	0.058706226
607	0.058800351
608	0.058894466
609	0.058988572
610	0.059082668
611	0.059176756
612	0.059270833
613	0.059364901
614	0.05945896
615	0.05955301
616	0.05964705
617	0.05974108
618	0.059835101
619	0.059929113
620	0.060023116
621	0.060117109
622	0.060211092
623	0.060305066
624	0.060399031
625	0.060492986
626	0.060586932
627	0.060680869
628	0.060774796
629	0.060868714
630	0.060962623
631	0.061056522
632	0.061150411
633	0.061244292
634	0.061338162
635	0.061432024
636	0.061525876
637	0.061619719
638	0.061713552
639	0.061807376
640	0.061901191
641	0.061994996
642	0.062088792
643	0.062182578
644	0.062276355
645	0.062370123
646	0.062463881
647	0.06255763
648	0.06265137
649	0.0627451
650	0.062838821
651	0.062932532
652	0.063026234
653	0.063119927
654	0.06321361
655	0.063307284
656	0.063400949
657	0.063494604
658	0.06358825
659	0.063681886
660	0.063775513
661	0.063869131
662	0.063962739
663	0.064056338
664	0.064149928
665	0.064243508
666	0.064337079
667	0.064430641
668	0.064524193
669	0.064617736
670	0.06471127
671	0.064804794
672	0.064898309
673	0.064991814
674	0.06508531
675	0.065178797
676	0.065272275
677	0.065365743
678	0.065459202
679	0.065552651
680	0.065646091
681	0.065739522
682	0.065832943
683	0.065926355
684	0.066019758
685	0.066113151
686	0.066206535
687	0.06629991
688	0.066393275
689	0.066486631
690	0.066579978
691	0.066673315
692	0.066766643
693	0.066859962
694	0.066953271
695	0.067046571
696	0.067139862
697	0.067233143
698	0.067326415
699	0.067419678
700	0.067512931
701	0.067606175
702	0.06769941
703	0.067792636
704	#NUM!

How do I avoid arbitrarily large exponents in calculations with such small answers? Is there a way of getting rid of them?

Last edited by a moderator:

You need to clarify your expression. You have several terms (100/100 = 1) which are confusing. Other terms are 100*100 = 10000. Why the product?

Modified:

kmarinas86 said:
$$y=\frac{\left((1-10000)e^{x/10000}+10000)e^{-x/10000}\right)-e^{-x}}{1-10000}$$

http://www.quickmath.com/msolver//graphs/2011-12-06/e5/61/e2/e561e26ac85a4b1da0176565a4827772-2.png?t=1323201139

Code:
0	0
1	3.67866E-05
2	0.000113525
3	0.000204954
4	0.000301782
5	0.000400589
6	0.000500118
7	0.000599906
8	0.000699784
9	0.000799687
10	0.000899595
11	0.000999497
12	0.001099391
13	0.001199276
14	0.00129915
15	0.001399015
16	0.001498871
17	0.001598716
18	0.001698551
19	0.001798376
20	0.001898191
21	0.001997996
22	0.002097792
23	0.002197577
24	0.002297352
25	0.002397117
26	0.002496873
27	0.002596618
28	0.002696353
29	0.002796079
30	0.002895794
31	0.0029955
32	0.003095195
33	0.00319488
34	0.003294556
35	0.003394222
36	0.003493877
37	0.003593523
38	0.003693158
39	0.003792784
40	0.0038924
41	0.003992006
42	0.004091601
43	0.004191187
44	0.004290763
45	0.004390329
46	0.004489885
47	0.004589431
48	0.004688967
49	0.004788493
50	0.00488801
51	0.004987516
52	0.005087012
53	0.005186498
54	0.005285975
55	0.005385441
56	0.005484898
57	0.005584344
58	0.005683781
59	0.005783208
60	0.005882624
61	0.005982031
62	0.006081428
63	0.006180815
64	0.006280192
65	0.006379559
66	0.006478916
67	0.006578263
68	0.0066776
69	0.006776927
70	0.006876245
71	0.006975552
72	0.00707485
73	0.007174137
74	0.007273415
75	0.007372682
76	0.00747194
77	0.007571188
78	0.007670426
79	0.007769654
80	0.007868872
81	0.00796808
82	0.008067278
83	0.008166467
84	0.008265645
85	0.008364814
86	0.008463972
87	0.008563121
88	0.00866226
89	0.008761388
90	0.008860507
91	0.008959616
92	0.009058715
93	0.009157805
94	0.009256884
95	0.009355953
96	0.009455013
97	0.009554062
98	0.009653102
99	0.009752132
100	0.009851151
101	0.009950161
102	0.010049161
103	0.010148151
104	0.010247132
105	0.010346102
106	0.010445062
107	0.010544013
108	0.010642954
109	0.010741884
110	0.010840805
111	0.010939716
112	0.011038617
113	0.011137509
114	0.01123639
115	0.011335261
116	0.011434123
117	0.011532974
118	0.011631816
119	0.011730648
120	0.01182947
121	0.011928282
122	0.012027084
123	0.012125877
124	0.012224659
125	0.012323432
126	0.012422195
127	0.012520947
128	0.01261969
129	0.012718423
130	0.012817147
131	0.01291586
132	0.013014564
133	0.013113257
134	0.013211941
135	0.013310615
136	0.013409279
137	0.013507933
138	0.013606577
139	0.013705212
140	0.013803836
141	0.013902451
142	0.014001056
143	0.014099651
144	0.014198236
145	0.014296811
146	0.014395376
147	0.014493932
148	0.014592478
149	0.014691013
150	0.014789539
151	0.014888055
152	0.014986562
153	0.015085058
154	0.015183545
155	0.015282021
156	0.015380488
157	0.015478945
158	0.015577393
159	0.01567583
160	0.015774257
161	0.015872675
162	0.015971083
163	0.016069481
164	0.016167869
165	0.016266247
166	0.016364616
167	0.016462974
168	0.016561323
169	0.016659662
170	0.016757991
171	0.01685631
172	0.01695462
173	0.01705292
174	0.017151209
175	0.017249489
176	0.017347759
177	0.01744602
178	0.01754427
179	0.017642511
180	0.017740742
181	0.017838963
182	0.017937174
183	0.018035375
184	0.018133567
185	0.018231749
186	0.01832992
187	0.018428083
188	0.018526235
189	0.018624377
190	0.01872251
191	0.018820633
192	0.018918746
193	0.019016849
194	0.019114943
195	0.019213026
196	0.0193111
197	0.019409164
198	0.019507218
199	0.019605262
200	0.019703297
201	0.019801322
202	0.019899337
203	0.019997342
204	0.020095337
205	0.020193323
206	0.020291299
207	0.020389265
208	0.020487221
209	0.020585167
210	0.020683104
211	0.020781031
212	0.020878948
213	0.020976855
214	0.021074752
215	0.02117264
216	0.021270518
217	0.021368386
218	0.021466244
219	0.021564092
220	0.021661931
221	0.02175976
222	0.021857579
223	0.021955389
224	0.022053188
225	0.022150978
226	0.022248758
227	0.022346528
228	0.022444289
229	0.022542039
230	0.02263978
231	0.022737511
232	0.022835233
233	0.022932944
234	0.023030646
235	0.023128338
236	0.02322602
237	0.023323693
238	0.023421356
239	0.023519009
240	0.023616652
241	0.023714285
242	0.023811909
243	0.023909523
244	0.024007127
245	0.024104722
246	0.024202306
247	0.024299881
248	0.024397446
249	0.024495002
250	0.024592547
251	0.024690083
252	0.024787609
253	0.024885126
254	0.024982632
255	0.025080129
256	0.025177616
257	0.025275094
258	0.025372561
259	0.025470019
260	0.025567467
261	0.025664906
262	0.025762334
263	0.025859753
264	0.025957162
265	0.026054562
266	0.026151951
267	0.026249331
268	0.026346701
269	0.026444062
270	0.026541413
271	0.026638754
272	0.026736085
273	0.026833406
274	0.026930718
275	0.02702802
276	0.027125313
277	0.027222595
278	0.027319868
279	0.027417131
280	0.027514385
281	0.027611628
282	0.027708862
283	0.027806087
284	0.027903301
285	0.028000506
286	0.028097701
287	0.028194886
288	0.028292062
289	0.028389228
290	0.028486384
291	0.028583531
292	0.028680667
293	0.028777795
294	0.028874912
295	0.02897202
296	0.029069118
297	0.029166206
298	0.029263284
299	0.029360353
300	0.029457412
301	0.029554462
302	0.029651501
303	0.029748531
304	0.029845552
305	0.029942562
306	0.030039563
307	0.030136554
308	0.030233536
309	0.030330508
310	0.03042747
311	0.030524422
312	0.030621365
313	0.030718298
314	0.030815221
315	0.030912135
316	0.031009039
317	0.031105933
318	0.031202818
319	0.031299692
320	0.031396558
321	0.031493413
322	0.031590259
323	0.031687095
324	0.031783921
325	0.031880738
326	0.031977545
327	0.032074343
328	0.03217113
329	0.032267909
330	0.032364677
331	0.032461436
332	0.032558185
333	0.032654924
334	0.032751654
335	0.032848374
336	0.032945084
337	0.033041785
338	0.033138476
339	0.033235157
340	0.033331829
341	0.033428491
342	0.033525143
343	0.033621786
344	0.033718418
345	0.033815042
346	0.033911655
347	0.034008259
348	0.034104854
349	0.034201439
350	0.034298014
351	0.034394579
352	0.034491135
353	0.034587681
354	0.034684217
355	0.034780744
356	0.034877261
357	0.034973768
358	0.035070266
359	0.035166754
360	0.035263233
361	0.035359702
362	0.035456161
363	0.03555261
364	0.03564905
365	0.035745481
366	0.035841901
367	0.035938312
368	0.036034714
369	0.036131105
370	0.036227487
371	0.03632386
372	0.036420223
373	0.036516576
374	0.036612919
375	0.036709253
376	0.036805577
377	0.036901892
378	0.036998197
379	0.037094492
380	0.037190778
381	0.037287054
382	0.037383321
383	0.037479578
384	0.037575825
385	0.037672062
386	0.03776829
387	0.037864509
388	0.037960718
389	0.038056917
390	0.038153106
391	0.038249286
392	0.038345456
393	0.038441617
394	0.038537768
395	0.038633909
396	0.038730041
397	0.038826163
398	0.038922276
399	0.039018379
400	0.039114472
401	0.039210556
402	0.03930663
403	0.039402695
404	0.03949875
405	0.039594795
406	0.039690831
407	0.039786857
408	0.039882873
409	0.03997888
410	0.040074878
411	0.040170865
412	0.040266843
413	0.040362812
414	0.040458771
415	0.04055472
416	0.04065066
417	0.04074659
418	0.040842511
419	0.040938421
420	0.041034323
421	0.041130215
422	0.041226097
423	0.041321969
424	0.041417832
425	0.041513686
426	0.04160953
427	0.041705364
428	0.041801189
429	0.041897004
430	0.041992809
431	0.042088605
432	0.042184391
433	0.042280168
434	0.042375935
435	0.042471693
436	0.042567441
437	0.04266318
438	0.042758908
439	0.042854628
440	0.042950338
441	0.043046038
442	0.043141728
443	0.043237409
444	0.043333081
445	0.043428743
446	0.043524395
447	0.043620038
448	0.043715671
449	0.043811295
450	0.043906909
451	0.044002513
452	0.044098108
453	0.044193694
454	0.04428927
455	0.044384836
456	0.044480393
457	0.04457594
458	0.044671477
459	0.044767006
460	0.044862524
461	0.044958033
462	0.045053532
463	0.045149022
464	0.045244503
465	0.045339973
466	0.045435435
467	0.045530886
468	0.045626328
469	0.045721761
470	0.045817184
471	0.045912598
472	0.046008002
473	0.046103396
474	0.046198781
475	0.046294156
476	0.046389522
477	0.046484878
478	0.046580225
479	0.046675562
480	0.04677089
481	0.046866208
482	0.046961517
483	0.047056816
484	0.047152105
485	0.047247385
486	0.047342656
487	0.047437917
488	0.047533168
489	0.04762841
490	0.047723643
491	0.047818866
492	0.047914079
493	0.048009283
494	0.048104477
495	0.048199662
496	0.048294837
497	0.048390003
498	0.048485159
499	0.048580306
500	0.048675443
501	0.048770571
502	0.048865689
503	0.048960798
504	0.049055897
505	0.049150986
506	0.049246067
507	0.049341137
508	0.049436198
509	0.04953125
510	0.049626292
511	0.049721325
512	0.049816348
513	0.049911361
514	0.050006366
515	0.05010136
516	0.050196345
517	0.050291321
518	0.050386287
519	0.050481244
520	0.050576191
521	0.050671128
522	0.050766057
523	0.050860975
524	0.050955884
525	0.051050784
526	0.051145674
527	0.051240555
528	0.051335426
529	0.051430288
530	0.05152514
531	0.051619983
532	0.051714816
533	0.05180964
534	0.051904454
535	0.051999259
536	0.052094054
537	0.05218884
538	0.052283616
539	0.052378383
540	0.052473141
541	0.052567889
542	0.052662627
543	0.052757356
544	0.052852076
545	0.052946786
546	0.053041486
547	0.053136178
548	0.053230859
549	0.053325531
550	0.053420194
551	0.053514847
552	0.053609491
553	0.053704125
554	0.05379875
555	0.053893366
556	0.053987972
557	0.054082568
558	0.054177155
559	0.054271733
560	0.054366301
561	0.054460859
562	0.054555409
563	0.054649948
564	0.054744479
565	0.054838999
566	0.054933511
567	0.055028013
568	0.055122505
569	0.055216988
570	0.055311462
571	0.055405926
572	0.055500381
573	0.055594826
574	0.055689262
575	0.055783688
576	0.055878105
577	0.055972512
578	0.05606691
579	0.056161299
580	0.056255678
581	0.056350048
582	0.056444408
583	0.056538759
584	0.0566331
585	0.056727432
586	0.056821755
587	0.056916068
588	0.057010372
589	0.057104666
590	0.057198951
591	0.057293226
592	0.057387492
593	0.057481749
594	0.057575996
595	0.057670233
596	0.057764462
597	0.057858681
598	0.05795289
599	0.05804709
600	0.058141281
601	0.058235462
602	0.058329633
603	0.058423796
604	0.058517949
605	0.058612092
606	0.058706226
607	0.058800351
608	0.058894466
609	0.058988572
610	0.059082668
611	0.059176756
612	0.059270833
613	0.059364901
614	0.05945896
615	0.05955301
616	0.05964705
617	0.05974108
618	0.059835101
619	0.059929113
620	0.060023116
621	0.060117109
622	0.060211092
623	0.060305066
624	0.060399031
625	0.060492986
626	0.060586932
627	0.060680869
628	0.060774796
629	0.060868714
630	0.060962623
631	0.061056522
632	0.061150411
633	0.061244292
634	0.061338162
635	0.061432024
636	0.061525876
637	0.061619719
638	0.061713552
639	0.061807376
640	0.061901191
641	0.061994996
642	0.062088792
643	0.062182578
644	0.062276355
645	0.062370123
646	0.062463881
647	0.06255763
648	0.06265137
649	0.0627451
650	0.062838821
651	0.062932532
652	0.063026234
653	0.063119927
654	0.06321361
655	0.063307284
656	0.063400949
657	0.063494604
658	0.06358825
659	0.063681886
660	0.063775513
661	0.063869131
662	0.063962739
663	0.064056338
664	0.064149928
665	0.064243508
666	0.064337079
667	0.064430641
668	0.064524193
669	0.064617736
670	0.06471127
671	0.064804794
672	0.064898309
673	0.064991814
674	0.06508531
675	0.065178797
676	0.065272275
677	0.065365743
678	0.065459202
679	0.065552651
680	0.065646091
681	0.065739522
682	0.065832943
683	0.065926355
684	0.066019758
685	0.066113151
686	0.066206535
687	0.06629991
688	0.066393275
689	0.066486631
690	0.066579978
691	0.066673315
692	0.066766643
693	0.066859962
694	0.066953271
695	0.067046571
696	0.067139862
697	0.067233143
698	0.067326415
699	0.067419678
700	0.067512931
701	0.067606175
702	0.06769941
703	0.067792636
704	#NUM!

How do I avoid arbitrarily large exponents in calculations with such small answers? Is there a way of getting rid of them?

Last edited by a moderator:

You could start by using the approximation eu = 1 + u + u2/2 for small u.

kmarinas86,

The parentheses aren't balanced in your modified version.

kmarinas86 said:
$$y=\frac{100}{100}\frac{\frac{(((100/100)^2-100*100)e^{x/(100*100)}+(100*100))e^{-x/(100*100)})-(100/100)^2e^{-x*100/100}}{100/100-100*100}}{1}$$

http://www.quickmath.com/msolver//graphs/2011-12-06/e5/61/e2/e561e26ac85a4b1da0176565a4827772-2.png?t=1323201139

Code:
0	0
1	3.67866E-05
2	0.000113525
3	0.000204954
4	0.000301782
5	0.000400589
6	0.000500118
7	0.000599906
8	0.000699784
9	0.000799687
10	0.000899595
11	0.000999497
12	0.001099391
13	0.001199276
14	0.00129915
15	0.001399015
16	0.001498871
17	0.001598716
18	0.001698551
19	0.001798376
20	0.001898191
21	0.001997996
22	0.002097792
23	0.002197577
24	0.002297352
25	0.002397117
26	0.002496873
27	0.002596618
28	0.002696353
29	0.002796079
30	0.002895794
31	0.0029955
32	0.003095195
33	0.00319488
34	0.003294556
35	0.003394222
36	0.003493877
37	0.003593523
38	0.003693158
39	0.003792784
40	0.0038924
41	0.003992006
42	0.004091601
43	0.004191187
44	0.004290763
45	0.004390329
46	0.004489885
47	0.004589431
48	0.004688967
49	0.004788493
50	0.00488801
51	0.004987516
52	0.005087012
53	0.005186498
54	0.005285975
55	0.005385441
56	0.005484898
57	0.005584344
58	0.005683781
59	0.005783208
60	0.005882624
61	0.005982031
62	0.006081428
63	0.006180815
64	0.006280192
65	0.006379559
66	0.006478916
67	0.006578263
68	0.0066776
69	0.006776927
70	0.006876245
71	0.006975552
72	0.00707485
73	0.007174137
74	0.007273415
75	0.007372682
76	0.00747194
77	0.007571188
78	0.007670426
79	0.007769654
80	0.007868872
81	0.00796808
82	0.008067278
83	0.008166467
84	0.008265645
85	0.008364814
86	0.008463972
87	0.008563121
88	0.00866226
89	0.008761388
90	0.008860507
91	0.008959616
92	0.009058715
93	0.009157805
94	0.009256884
95	0.009355953
96	0.009455013
97	0.009554062
98	0.009653102
99	0.009752132
100	0.009851151
101	0.009950161
102	0.010049161
103	0.010148151
104	0.010247132
105	0.010346102
106	0.010445062
107	0.010544013
108	0.010642954
109	0.010741884
110	0.010840805
111	0.010939716
112	0.011038617
113	0.011137509
114	0.01123639
115	0.011335261
116	0.011434123
117	0.011532974
118	0.011631816
119	0.011730648
120	0.01182947
121	0.011928282
122	0.012027084
123	0.012125877
124	0.012224659
125	0.012323432
126	0.012422195
127	0.012520947
128	0.01261969
129	0.012718423
130	0.012817147
131	0.01291586
132	0.013014564
133	0.013113257
134	0.013211941
135	0.013310615
136	0.013409279
137	0.013507933
138	0.013606577
139	0.013705212
140	0.013803836
141	0.013902451
142	0.014001056
143	0.014099651
144	0.014198236
145	0.014296811
146	0.014395376
147	0.014493932
148	0.014592478
149	0.014691013
150	0.014789539
151	0.014888055
152	0.014986562
153	0.015085058
154	0.015183545
155	0.015282021
156	0.015380488
157	0.015478945
158	0.015577393
159	0.01567583
160	0.015774257
161	0.015872675
162	0.015971083
163	0.016069481
164	0.016167869
165	0.016266247
166	0.016364616
167	0.016462974
168	0.016561323
169	0.016659662
170	0.016757991
171	0.01685631
172	0.01695462
173	0.01705292
174	0.017151209
175	0.017249489
176	0.017347759
177	0.01744602
178	0.01754427
179	0.017642511
180	0.017740742
181	0.017838963
182	0.017937174
183	0.018035375
184	0.018133567
185	0.018231749
186	0.01832992
187	0.018428083
188	0.018526235
189	0.018624377
190	0.01872251
191	0.018820633
192	0.018918746
193	0.019016849
194	0.019114943
195	0.019213026
196	0.0193111
197	0.019409164
198	0.019507218
199	0.019605262
200	0.019703297
201	0.019801322
202	0.019899337
203	0.019997342
204	0.020095337
205	0.020193323
206	0.020291299
207	0.020389265
208	0.020487221
209	0.020585167
210	0.020683104
211	0.020781031
212	0.020878948
213	0.020976855
214	0.021074752
215	0.02117264
216	0.021270518
217	0.021368386
218	0.021466244
219	0.021564092
220	0.021661931
221	0.02175976
222	0.021857579
223	0.021955389
224	0.022053188
225	0.022150978
226	0.022248758
227	0.022346528
228	0.022444289
229	0.022542039
230	0.02263978
231	0.022737511
232	0.022835233
233	0.022932944
234	0.023030646
235	0.023128338
236	0.02322602
237	0.023323693
238	0.023421356
239	0.023519009
240	0.023616652
241	0.023714285
242	0.023811909
243	0.023909523
244	0.024007127
245	0.024104722
246	0.024202306
247	0.024299881
248	0.024397446
249	0.024495002
250	0.024592547
251	0.024690083
252	0.024787609
253	0.024885126
254	0.024982632
255	0.025080129
256	0.025177616
257	0.025275094
258	0.025372561
259	0.025470019
260	0.025567467
261	0.025664906
262	0.025762334
263	0.025859753
264	0.025957162
265	0.026054562
266	0.026151951
267	0.026249331
268	0.026346701
269	0.026444062
270	0.026541413
271	0.026638754
272	0.026736085
273	0.026833406
274	0.026930718
275	0.02702802
276	0.027125313
277	0.027222595
278	0.027319868
279	0.027417131
280	0.027514385
281	0.027611628
282	0.027708862
283	0.027806087
284	0.027903301
285	0.028000506
286	0.028097701
287	0.028194886
288	0.028292062
289	0.028389228
290	0.028486384
291	0.028583531
292	0.028680667
293	0.028777795
294	0.028874912
295	0.02897202
296	0.029069118
297	0.029166206
298	0.029263284
299	0.029360353
300	0.029457412
301	0.029554462
302	0.029651501
303	0.029748531
304	0.029845552
305	0.029942562
306	0.030039563
307	0.030136554
308	0.030233536
309	0.030330508
310	0.03042747
311	0.030524422
312	0.030621365
313	0.030718298
314	0.030815221
315	0.030912135
316	0.031009039
317	0.031105933
318	0.031202818
319	0.031299692
320	0.031396558
321	0.031493413
322	0.031590259
323	0.031687095
324	0.031783921
325	0.031880738
326	0.031977545
327	0.032074343
328	0.03217113
329	0.032267909
330	0.032364677
331	0.032461436
332	0.032558185
333	0.032654924
334	0.032751654
335	0.032848374
336	0.032945084
337	0.033041785
338	0.033138476
339	0.033235157
340	0.033331829
341	0.033428491
342	0.033525143
343	0.033621786
344	0.033718418
345	0.033815042
346	0.033911655
347	0.034008259
348	0.034104854
349	0.034201439
350	0.034298014
351	0.034394579
352	0.034491135
353	0.034587681
354	0.034684217
355	0.034780744
356	0.034877261
357	0.034973768
358	0.035070266
359	0.035166754
360	0.035263233
361	0.035359702
362	0.035456161
363	0.03555261
364	0.03564905
365	0.035745481
366	0.035841901
367	0.035938312
368	0.036034714
369	0.036131105
370	0.036227487
371	0.03632386
372	0.036420223
373	0.036516576
374	0.036612919
375	0.036709253
376	0.036805577
377	0.036901892
378	0.036998197
379	0.037094492
380	0.037190778
381	0.037287054
382	0.037383321
383	0.037479578
384	0.037575825
385	0.037672062
386	0.03776829
387	0.037864509
388	0.037960718
389	0.038056917
390	0.038153106
391	0.038249286
392	0.038345456
393	0.038441617
394	0.038537768
395	0.038633909
396	0.038730041
397	0.038826163
398	0.038922276
399	0.039018379
400	0.039114472
401	0.039210556
402	0.03930663
403	0.039402695
404	0.03949875
405	0.039594795
406	0.039690831
407	0.039786857
408	0.039882873
409	0.03997888
410	0.040074878
411	0.040170865
412	0.040266843
413	0.040362812
414	0.040458771
415	0.04055472
416	0.04065066
417	0.04074659
418	0.040842511
419	0.040938421
420	0.041034323
421	0.041130215
422	0.041226097
423	0.041321969
424	0.041417832
425	0.041513686
426	0.04160953
427	0.041705364
428	0.041801189
429	0.041897004
430	0.041992809
431	0.042088605
432	0.042184391
433	0.042280168
434	0.042375935
435	0.042471693
436	0.042567441
437	0.04266318
438	0.042758908
439	0.042854628
440	0.042950338
441	0.043046038
442	0.043141728
443	0.043237409
444	0.043333081
445	0.043428743
446	0.043524395
447	0.043620038
448	0.043715671
449	0.043811295
450	0.043906909
451	0.044002513
452	0.044098108
453	0.044193694
454	0.04428927
455	0.044384836
456	0.044480393
457	0.04457594
458	0.044671477
459	0.044767006
460	0.044862524
461	0.044958033
462	0.045053532
463	0.045149022
464	0.045244503
465	0.045339973
466	0.045435435
467	0.045530886
468	0.045626328
469	0.045721761
470	0.045817184
471	0.045912598
472	0.046008002
473	0.046103396
474	0.046198781
475	0.046294156
476	0.046389522
477	0.046484878
478	0.046580225
479	0.046675562
480	0.04677089
481	0.046866208
482	0.046961517
483	0.047056816
484	0.047152105
485	0.047247385
486	0.047342656
487	0.047437917
488	0.047533168
489	0.04762841
490	0.047723643
491	0.047818866
492	0.047914079
493	0.048009283
494	0.048104477
495	0.048199662
496	0.048294837
497	0.048390003
498	0.048485159
499	0.048580306
500	0.048675443
501	0.048770571
502	0.048865689
503	0.048960798
504	0.049055897
505	0.049150986
506	0.049246067
507	0.049341137
508	0.049436198
509	0.04953125
510	0.049626292
511	0.049721325
512	0.049816348
513	0.049911361
514	0.050006366
515	0.05010136
516	0.050196345
517	0.050291321
518	0.050386287
519	0.050481244
520	0.050576191
521	0.050671128
522	0.050766057
523	0.050860975
524	0.050955884
525	0.051050784
526	0.051145674
527	0.051240555
528	0.051335426
529	0.051430288
530	0.05152514
531	0.051619983
532	0.051714816
533	0.05180964
534	0.051904454
535	0.051999259
536	0.052094054
537	0.05218884
538	0.052283616
539	0.052378383
540	0.052473141
541	0.052567889
542	0.052662627
543	0.052757356
544	0.052852076
545	0.052946786
546	0.053041486
547	0.053136178
548	0.053230859
549	0.053325531
550	0.053420194
551	0.053514847
552	0.053609491
553	0.053704125
554	0.05379875
555	0.053893366
556	0.053987972
557	0.054082568
558	0.054177155
559	0.054271733
560	0.054366301
561	0.054460859
562	0.054555409
563	0.054649948
564	0.054744479
565	0.054838999
566	0.054933511
567	0.055028013
568	0.055122505
569	0.055216988
570	0.055311462
571	0.055405926
572	0.055500381
573	0.055594826
574	0.055689262
575	0.055783688
576	0.055878105
577	0.055972512
578	0.05606691
579	0.056161299
580	0.056255678
581	0.056350048
582	0.056444408
583	0.056538759
584	0.0566331
585	0.056727432
586	0.056821755
587	0.056916068
588	0.057010372
589	0.057104666
590	0.057198951
591	0.057293226
592	0.057387492
593	0.057481749
594	0.057575996
595	0.057670233
596	0.057764462
597	0.057858681
598	0.05795289
599	0.05804709
600	0.058141281
601	0.058235462
602	0.058329633
603	0.058423796
604	0.058517949
605	0.058612092
606	0.058706226
607	0.058800351
608	0.058894466
609	0.058988572
610	0.059082668
611	0.059176756
612	0.059270833
613	0.059364901
614	0.05945896
615	0.05955301
616	0.05964705
617	0.05974108
618	0.059835101
619	0.059929113
620	0.060023116
621	0.060117109
622	0.060211092
623	0.060305066
624	0.060399031
625	0.060492986
626	0.060586932
627	0.060680869
628	0.060774796
629	0.060868714
630	0.060962623
631	0.061056522
632	0.061150411
633	0.061244292
634	0.061338162
635	0.061432024
636	0.061525876
637	0.061619719
638	0.061713552
639	0.061807376
640	0.061901191
641	0.061994996
642	0.062088792
643	0.062182578
644	0.062276355
645	0.062370123
646	0.062463881
647	0.06255763
648	0.06265137
649	0.0627451
650	0.062838821
651	0.062932532
652	0.063026234
653	0.063119927
654	0.06321361
655	0.063307284
656	0.063400949
657	0.063494604
658	0.06358825
659	0.063681886
660	0.063775513
661	0.063869131
662	0.063962739
663	0.064056338
664	0.064149928
665	0.064243508
666	0.064337079
667	0.064430641
668	0.064524193
669	0.064617736
670	0.06471127
671	0.064804794
672	0.064898309
673	0.064991814
674	0.06508531
675	0.065178797
676	0.065272275
677	0.065365743
678	0.065459202
679	0.065552651
680	0.065646091
681	0.065739522
682	0.065832943
683	0.065926355
684	0.066019758
685	0.066113151
686	0.066206535
687	0.06629991
688	0.066393275
689	0.066486631
690	0.066579978
691	0.066673315
692	0.066766643
693	0.066859962
694	0.066953271
695	0.067046571
696	0.067139862
697	0.067233143
698	0.067326415
699	0.067419678
700	0.067512931
701	0.067606175
702	0.06769941
703	0.067792636
704	#NUM!

How do I avoid arbitrarily large exponents in calculations with such small answers? Is there a way of getting rid of them?

It appears that this was easier than I thought:

$$y=\frac{100}{100}\frac{\frac{((((100/100)^2-100*100)e^{x/(100*100)}+(100*100))e^{-x/(100*100)})-(100/100)^2e^{-x*100/100}}{100/100-100*100}}{1}$$

$$y=\frac{100}{100}\frac{((((100/100)^2-100*100)e^{x/(100*100)}+(100*100))e^{-x/(100*100)})-(100/100)^2e^{-x*100/100}}{100/100-100*100}$$

$$y=\frac{(((1-10000)e^{x/10000}+10000)e^{-x/10000})-e^{-x}}{1-10000}$$

$$y=\frac{(((1-10000)e^{x/10000}e^{-x/10000})+10000e^{-x/10000}))-e^{-x}}{1-10000}$$

$$1=e^{x}e^{-x}$$

$$y=\frac{1-10000+10000e^{-x/10000}-e^{-x}}{1-10000}$$

I figured this out a few days ago, but I hadn't decided to post it until now.

Last edited by a moderator:

## 1. How does avoiding arbitrarily large exponents affect the accuracy of calculations?

Avoiding arbitrarily large exponents can improve the accuracy of calculations by preventing overflow errors and loss of precision. This is especially important when working with small numbers, as even slight rounding errors can significantly impact the final result.

## 2. What techniques can be used to systematically avoid arbitrary large exponents?

One technique is to break down the calculation into smaller steps, using intermediate results that are within a manageable range. Another approach is to use scientific notation, which allows for a more compact representation of numbers with large or small exponents.

## 3. Why is it important to avoid arbitrarily large exponents in scientific calculations?

In scientific calculations, accuracy is crucial. Using arbitrarily large exponents can lead to errors and inaccuracies, which can impact the validity and reliability of research findings. It is important to use precise and consistent methods to ensure accurate results.

## 4. Can avoiding arbitrarily large exponents also affect the efficiency of calculations?

Yes, avoiding arbitrarily large exponents can also improve the efficiency of calculations. By breaking down the calculation into smaller steps, the overall computational load is reduced, making it faster and more efficient. Additionally, using scientific notation can also save time and memory space.

## 5. How can one ensure that arbitrarily large exponents are consistently avoided in calculations?

One way to ensure consistent avoidance of arbitrarily large exponents is to establish clear guidelines and protocols for handling calculations with small answers. This can include using scientific notation, setting limits for exponent values, and regularly checking for and correcting any potential errors.