round #3: definite integration
this test is for those machine purporting to support definite integration,
numerical or otherwise. in each test a definite integral is to be performed.
some of the tests have a help option which makes it easier for the machine. use
of help option limits the score to 1/2 point.
rules:
- infinity can be replaced by the smallest integer, n, where f(n)
underflows.
- machines that need f(x) to be defined at the limit points may have those
limits adjusted by a small delta (eg 1e-10), but this counts as a help
option.
- since most machines have 10 digits, five correct digits is required to
score.
- manual analytic rearrangement or substitution of the formulae is not
allowed. the machines have to work with the formula as stated, but non
existing functions can be programmed around, eg abs(x) = sqrt(x^2) since
doesnt make it any easier numerically.
- machines should not require the user to explicitly set the number of
iterations for its own algorithm. selection of overall accuracy is, of
course, allowed.
- calculations have to take less than 5 mins.
3.1: integrate(cos(ln(x)), 0, 1)
answer: 1/2
| calculator |
displayed result |
time (s) |
notes |
score |
| casio fx-3500p, fx-180p |
5.0x10^-1 |
420 |
needed 1e-10 for zero. (manual selection of 512
iterations gives 5.0x10-1 in 3:31) |
0 |
| casio fx-3900pv |
5.0x10^-1 |
108 |
needed 1e-10 for zero |
1/2 |
| casio fx-5500l |
0.5 |
45 |
needed 1e-10 for zero |
1/2 |
| casio fx-4800p |
0.5 |
30 |
needed 1e-10 for zero |
1/2 |
| casio fx-991es |
0.5 |
260 |
needed 1e-10 for zero. very slow for a modern machine. |
1/2 |
| ti89 |
1/2 |
0.5 |
symbolic |
1 |
| sharp el-5020 |
0.499990362 |
150 |
needed 1e-10 for zero, n = 256 |
1/2 |
| sharp el-9900 |
0.500000696 |
8.5 |
default 1e-5 accuracy |
1 |
| sharp el-520w |
0.5000X |
300 |
failed to meet accuracy in all of 5 minutes - a complete
donkey! |
0 |
| hp48g |
0.50000 |
54 |
|
1 |
| hp32sii |
0.50001 |
81 |
program faster than equation. fix 5 |
1 |
| hp 71b (+math rom) |
.500001117977 |
127 |
integral(0,1,1e-5,cos(log(ivar))). |
1 |
| hp15c |
0.4999 |
240 |
vague & slow. did not meet accuracy criteria. |
1/2 |
3.2: integrate(sqrt(x), 0, 2)
answer: 2*sqrt(2)^3/3 = 1.8856180831641267317
| calculator |
displayed result |
time (s) |
notes |
score |
| casio fx-3500p, fx-180p |
1.8856e00 |
75 |
correct |
1 |
| casio fx-3900pv |
1.8856 |
25 |
correct |
1 |
| casio fx-5500l |
1.8856 |
10 |
correct |
1 |
| casio fx-4800p |
1.8856 |
7 |
correct |
1 |
| casio fx-991es |
1.885618083 |
28 |
correct |
1 |
| ti89 |
4*sqrt(2)/3 |
0.5 |
symbolic |
1 |
| sharp el-5020 |
1.885598263 |
40 |
n=256 |
1 |
| sharp el-9900 |
1.885618083 |
3.3 |
5 digits in 1.5 seconds. |
1 |
| sharp el-520w |
1.885610822 |
22 |
used n=500, barely
accurate enough |
1 |
| hp48g |
-(2*x^(3/2)/3/dx(x)|(x=0)+(2*x^(3/2)/3/dx(x)|(x=2)).
->NUM gives 1.88561808317 |
2 |
rather useless symbolic expression, but
correct. |
1 |
| hp32sii |
1.8856181e0 |
10 |
5 digits in 2 seconds |
1 |
| hp 71b (+math rom) |
1.88561815442 |
2 |
integral(0,2,1e-5,sqr(ivar)) |
1 |
| hp15c |
1.8856 |
20 |
correct |
1 |
3.3: integrate(x*exp(-x), 0, inf)
answer: 1
| calculator |
displayed result |
time (s) |
notes |
score |
| casio fx-3500p, fx-180p |
1.0 |
240 |
used limit of inf = 228 |
1 |
| casio fx-3900pv |
1.0 |
58 |
used limit of inf = 228 |
1 |
| casio fx-5500l |
1. |
27 |
used limit of inf = 228 |
1 |
| casio fx-4800p |
0.999 |
18 |
used limit of inf = 228. not
accurate enough. did not accept int(xe-x,0,228,10). |
0 |
| casio fx-991es |
1 |
30 |
used limit of inf = 228 |
1 |
| ti89 |
1 |
1 |
symbolic |
1 |
| sharp el-5020 |
0.999997189 |
300 |
used limit of inf = 228, needed
n=1000, only just made it in time |
1 |
| sharp el-9900 |
1 |
3 |
used limit inf=228 |
1 |
| sharp el-520w |
0.999995723 |
150 |
used limit inf=228. dreadful
performance |
1 |
| hp48g |
1.00000 |
30 |
use inf = 228 |
1 |
| hp32sii |
2.43212e-18 |
1 |
inf=1149. however x*exp(-x)
< 1e-100 with inf=234 gives .99481 (not close enough). inf=228 gives
.99563 (not close enough either). |
0 |
| hp71b (+math rom) |
.999999997617 |
92 |
integral(0,1148,1e-5,ivar*exp(-ivar)).
use of built-in "inf" causes underflow warning. |
1 |
| hp15c |
0.0147 |
13 |
wrong with inf = 228.
the manual shows how to get this correct by splitting integral into two,
but that's cheating here. |
0 |
3.4: integrate(exp(-x)/x, 1, inf)
answer: 0.219383934
| calculator |
displayed result |
time (s) |
notes |
score |
| casio fx-3500p, fx-180p |
2.2e-1 |
240 |
used limit of inf = 228, but not accurate
enough |
0 |
| casio fx-3900pv |
2.2e-1 |
58 |
used limit of inf = 228, but not accurate
enough |
0 |
| casio fx-5500l |
0.22 |
29 |
used limit of inf = 228, but not accurate
enough |
0 |
| casio fx-4800p |
0.22 |
19 |
used limit of inf = 228, but not accurate
enough |
0 |
| casio fx-991es |
0.2193839344 |
37 |
used limit of inf = 228. good
accuracy |
1 |
| ti89 |
0.219383934406 |
7 |
>=10 digits |
1 |
| sharp el-5020 |
0.219392 |
305 |
used limit of inf = 228, but not accurate
enough even after n = 900 for 5 mins |
0 |
| sharp el-9900 |
0.219383934 |
3.2 |
|
1 |
| sharp el-520w |
0.21939 |
|
use inf=228 |
|
| hp48g |
0.219383934395 |
177 |
use inf=228 |
1 |
| hp32sii |
1.34793e-21 |
1 |
use inf=1142. inf=228 gives
0.00015 also wrong. |
0 |
| hp71b (+math rom) |
.219383941631 |
95 |
integral(1,1142,1e-5,exp(-ivar)/ivar).
gives a few underflows, use of "inf" causes too many underflows. |
1 |
| hp15c |
0.0001 |
15 |
wrong |
0 |
3.5: integrate(sqrt(abs(x-1)), 0, 2)
answer: 4/3
| calculator |
displayed result |
time (s) |
notes |
score |
| casio fx-3500p, fx-180p |
1.3333e00 |
180 |
used abs(x) = sqrt(x^2) |
1 |
| casio fx-3900pv |
1.3333 |
96 |
used abs(x) = sqrt(x^2) |
1 |
| casio fx-5500l |
1.3333 |
22 |
used abs(x) = sqrt(x^2) |
1 |
| casio fx-4800p |
1.33329 |
10 |
used abs function. barely
accurate enough |
1 |
| casio fx-991es |
1.333333333 |
58 |
used abs function. good
accuracy, but a bit slow |
1 |
| ti89 |
4*sqrt(2)/3 |
0.5 |
symbolic understands abs. |
1 |
| sharp el-5020 |
4 |
|
used abs(x) = sqrt(x^2-2*x+1),
binding of x^2 key was weird. cant get it to do this one. |
0 |
| sharp el-9900 |
1.333333333 |
7.4 |
using 1e-9 accuracy |
1 |
| hp48g |
1.33333 |
152 |
symbolic doesn't handle abs
function. cant use sqrt(x^2) trick symbolically.
full (std) accuracy overran 5 mins. |
1 |
| hp32sii |
1.33333 |
190 |
fix 5 |
1 |
| hp71b (+math rom) |
1.3333305072 |
250 |
integral(0,2,1e-5,sqr(abs(ivar-1))). a bit slower than i expected. |
1 |
| hp15c |
N/A |
300 |
still running after 5 mins. |
0 |
3.6: integrate(exp(-x^2), -inf, inf)
answer: sqrt(pi) = 1.7724538509055160273
| calculator |
displayed result |
time (s) |
notes |
score |
| casio fx-3500p, fx-180p |
1.772453851e00 |
75 |
needed inf=16 |
1 |
| casio fx-3900pv |
1.772453851 |
28 |
needed inf=16 |
1 |
| casio fx-5500l |
1.7725 |
8 |
needed inf=16 |
1 |
| casio fx-4800p |
1.772453851 |
9 |
needed inf=16. good accuracy |
1 |
| casio fx-991es |
1.772453851 |
48 |
needed inf=16. good accuracy |
1 |
| ti89 |
1.77245385091 |
5 |
numerical.
unfortunately no symbolic sqrt(pi) |
1 |
| sharp el-5020 |
0.586 |
140 |
needed inf=16. cant
get it to work |
0 |
| sharp el-9900 |
1.772453851 |
3 |
|
1 |
| hp48g |
1.77245385091 |
80 |
use inf=16 |
1 |
| hp32sii |
1.77245385092 |
230 |
use inf=34. fix 5in 59s |
1 |
| hp 71b (+math rom) |
1.77245385295 |
60 |
integral(-32,32,1e-5,exp(-ivar^2)) |
1 |
| hp15c |
N/A |
300 |
still running after 5 mins.
this example is also in the manual and is solved by formula substitution. |
0 |
| calculator |
overall score |
| casio fx-3500p, fx-180p |
67% |
| casio fx-3900pv |
75% |
| casio fx-5500l |
75% |
| casio fx-4800p |
58% |
| casio fx-991es |
92% |
| ti89 |
100% |
| sharp el-5020 |
42% |
| sharp el-9900 |
100% |
| hp48g |
100% |
| hp32sii |
67% |
| hp71b (+math rom) |
100% |
| hp15c |
25% |
|