What could be the reason for this fzero error and how to solve it ?
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This is the error message
This was the warning I got too along with the error message
This is the 5 lines of code to reproduce the error
dd = 0.1069;
vv = 0.7889;
L = 1;
fun = @(aa) ( (-(sqrt(L^2-(vv)^2)/2)) + (aa*sinh(((dd)/2)/aa)) ); xx0 = 0.15; %[10^-6, 10^6];
aa = abs(fzero(fun,xx0));
13 个评论
Jon
2024-2-2
Please provide a self contained, runnable example of code the reproduces the problem so we can see exactly what is going wrong
Dyuman Joshi
2024-2-2
Plotting a function is always helpful, so let's do that. The function looks symmetric.
As aa approaches 0, the function approaches Infinity fast. And by fast, I mean like fast and furious movies produced without any thought or effort. (For legal reasons this is just a joke)
dd = 0.1069;
vv = 0.7889;
L = 1;
fun = @(aa) ( (-(sqrt(L^2-(vv)^2)/2)) + (aa.*sinh(((dd)/2)./aa)) );
fplot(fun)
Using a smaller value for the initial guess works. Well, it works not because the value is smaller, but it is closer to the solutions.
out1 = fzero(fun,0.005)
out1 = 0.0142
aa1 = abs(out1)
aa1 = 0.0142
Note that there are 2 solutions for the equation.
out2 = fzero(fun,-0.005)
out2 = -0.0142
aa1 = abs(out2)
aa1 = 0.0142
Understanding how fzero works is important. For a scalar initial value - fzero begins at x0 and tries to locate a point x1 where fun(x1) has the opposite sign of fun(x0). Then fzero iteratively shrinks the interval where fun changes sign to reach a solution.
If you start of the wrong foot, you are going to drown in the pool of weirdness of the mathematics followed.
AA = fsolve(fun, 0.5)
Equation solved.
fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.
AA = 0.0142
Let's see each iteration of fzero for the initial point 0.15.
Keep scrolling a bit and you will find the "weirdness" I talked about.
options = optimset('Display','iter'); % show iterations
[x, fval, exitflag] = fzero(fun, 0.15, options)
Search for an interval around 0.15 containing a sign change:
Func-count a f(a) b f(b) Procedure
1 0.15 -0.252672 0.15 -0.252672 initial interval
3 0.145757 -0.252605 0.154243 -0.252735 search
5 0.144 -0.252575 0.156 -0.252759 search
7 0.141515 -0.252531 0.158485 -0.252792 search
9 0.138 -0.252464 0.162 -0.252836 search
11 0.133029 -0.252361 0.166971 -0.252893 search
13 0.126 -0.252193 0.174 -0.252966 search
15 0.116059 -0.251901 0.183941 -0.253055 search
17 0.102 -0.251331 0.198 -0.253159 search
19 0.0821177 -0.249956 0.217882 -0.253273 search
21 0.054 -0.244645 0.246 -0.253389 search
23 0.0142355 -0.00334771 0.285765 -0.253499 search
25 -0.042 -0.238169 0.342 -0.253593 search
27 -0.121529 -0.252071 0.421529 -0.253667 search
29 -0.234 -0.253345 0.534 -0.253722 search
31 -0.393058 -0.253646 0.693058 -0.253758 search
33 -0.618 -0.253744 0.918 -0.253781 search
35 -0.936116 -0.253782 1.23612 -0.253794 search
37 -1.386 -0.253798 1.686 -0.253802 search
39 -2.02223 -0.253805 2.32223 -0.253806 search
41 -2.922 -0.253808 3.222 -0.253808 search
43 -4.19446 -0.253809 4.49446 -0.25381 search
45 -5.994 -0.25381 6.294 -0.25381 search
47 -8.53893 -0.25381 8.83893 -0.25381 search
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57 -49.002 -0.253811 49.302 -0.253811 search
59 -69.3614 -0.253811 69.6614 -0.253811 search
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121 -3.22123e+06 -0.253811 3.22123e+06 -0.253811 search
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1527 -2.08961e+112 -0.253811 2.08961e+112 -0.253811 search
1529 -2.95515e+112 -0.253811 2.95515e+112 -0.253811 search
1531 -4.17921e+112 -0.253811 4.17921e+112 -0.253811 search
1533 -5.9103e+112 -0.253811 5.9103e+112 -0.253811 search
1535 -8.35843e+112 -0.253811 8.35843e+112 -0.253811 search
1537 -1.18206e+113 -0.253811 1.18206e+113 -0.253811 search
1539 -1.67169e+113 -0.253811 1.67169e+113 -0.253811 search
1541 -2.36412e+113 -0.253811 2.36412e+113 -0.253811 search
1543 -3.34337e+113 -0.253811 3.34337e+113 -0.253811 search
1545 -4.72824e+113 -0.253811 4.72824e+113 -0.253811 search
1547 -6.68674e+113 -0.253811 6.68674e+113 -0.253811 search
1549 -9.45648e+113 -0.253811 9.45648e+113 -0.253811 search
1551 -1.33735e+114 -0.253811 1.33735e+114 -0.253811 search
1553 -1.8913e+114 -0.253811 1.8913e+114 -0.253811 search
1555 -2.6747e+114 -0.253811 2.6747e+114 -0.253811 search
1557 -3.78259e+114 -0.253811 3.78259e+114 -0.253811 search
1559 -5.34939e+114 -0.253811 5.34939e+114 -0.253811 search
1561 -7.56519e+114 -0.253811 7.56519e+114 -0.253811 search
1563 -1.06988e+115 -0.253811 1.06988e+115 -0.253811 search
1565 -1.51304e+115 -0.253811 1.51304e+115 -0.253811 search
1567 -2.13976e+115 -0.253811 2.13976e+115 -0.253811 search
1569 -3.02607e+115 -0.253811 3.02607e+115 -0.253811 search
1571 -4.27951e+115 -0.253811 4.27951e+115 -0.253811 search
1573 -6.05215e+115 -0.253811 6.05215e+115 -0.253811 search
1575 -8.55903e+115 -0.253811 8.55903e+115 -0.253811 search
1577 -1.21043e+116 -0.253811 1.21043e+116 -0.253811 search
1579 -1.71181e+116 -0.253811 1.71181e+116 -0.253811 search
1581 -2.42086e+116 -0.253811 2.42086e+116 -0.253811 search
1583 -3.42361e+116 -0.253811 3.42361e+116 -0.253811 search
1585 -4.84172e+116 -0.253811 4.84172e+116 -0.253811 search
1587 -6.84722e+116 -0.253811 6.84722e+116 -0.253811 search
1589 -9.68344e+116 -0.253811 9.68344e+116 -0.253811 search
1591 -1.36944e+117 -0.253811 1.36944e+117 -0.253811 search
1593 -1.93669e+117 -0.253811 1.93669e+117 -0.253811 search
1595 -2.73889e+117 -0.253811 2.73889e+117 -0.253811 search
1597 -3.87337e+117 -0.253811 3.87337e+117 -0.253811 search
1599 -5.47778e+117 -0.253811 5.47778e+117 -0.253811 search
1601 -7.74675e+117 -0.253811 7.74675e+117 -0.253811 search
1603 -1.09556e+118 -0.253811 1.09556e+118 -0.253811 search
1605 -1.54935e+118 -0.253811 1.54935e+118 -0.253811 search
1607 -2.19111e+118 -0.253811 2.19111e+118 -0.253811 search
1609 -3.0987e+118 -0.253811 3.0987e+118 -0.253811 search
1611 -4.38222e+118 -0.253811 4.38222e+118 -0.253811 search
1613 -6.1974e+118 -0.253811 6.1974e+118 -0.253811 search
1615 -8.76445e+118 -0.253811 8.76445e+118 -0.253811 search
1617 -1.23948e+119 -0.253811 1.23948e+119 -0.253811 search
1619 -1.75289e+119 -0.253811 1.75289e+119 -0.253811 search
1621 -2.47896e+119 -0.253811 2.47896e+119 -0.253811 search
1623 -3.50578e+119 -0.253811 3.50578e+119 -0.253811 search
1625 -4.95792e+119 -0.253811 4.95792e+119 -0.253811 search
1627 -7.01156e+119 -0.253811 7.01156e+119 -0.253811 search
1629 -9.91584e+119 -0.253811 9.91584e+119 -0.253811 search
1631 -1.40231e+120 -0.253811 1.40231e+120 -0.253811 search
1633 -1.98317e+120 -0.253811 1.98317e+120 -0.253811 search
1635 -2.80462e+120 -0.253811 2.80462e+120 -0.253811 search
1637 -3.96634e+120 -0.253811 3.96634e+120 -0.253811 search
1639 -5.60925e+120 -0.253811 5.60925e+120 -0.253811 ...
x = NaN
fval = NaN
exitflag = -3
VIGNESH BALAJI
2024-2-3
@Dyuman Joshi Thanks a lot for the awesome and detailed insight you gave me and on the approach of the problem. You appraoch and the way you explained it is very detailed and insightful to make a decision.
I am unable to undertsand the principle of operation of fzero and folve. I get a basic understanding that fzero solves the non linear function starting from a given initial guess and it keeps iterating values up and down around that region to look for a value where the non linear function changes it's sign. If the styarting initial guess is not close to the actual value the it is destined not to find the answer and return a NaN.
I have few questions in this ?
- What is the maximum search region in fzero and how it samples the search region ?
When I see the fzero iterations you showed me, I see that the search region passes over the region of the answer from iteration 23 to 25. I get a thought that maybe the sampling inside the search region is uneven and when I see it carefully, I see that the distance between samples gradually grows as the iteration happens.
2. What is the working principle of fzero ?
I see that it works on bisection method. I need to understand it a little more
3. How does fsolve differ from fzero ?
I understand that fsolve does not rely on sign change of the non linear function starting from the search region. It also suses a start region for search. I guess it looks for a zero touch than the zero crossing. I am in the process of understanding this deeper.
Please let me know your views and suggestions on my 3 questions. I am parallely reading the documentation to understand them.
Torsten
2024-2-3
编辑:Torsten
2024-2-3
What is the maximum search region in fzero and how it samples the search region ?
This comes from octave's "fzero". Edit the MATLAB source code and search for a similar line:
## Search in an ever-widening range around the initial point.
207 for srch = [-.01 +.025 -.05 +.10 -.25 +.50 -1 +2.5 -5 +10 -50 +100 -500 +1000]
What is the working principle of fzero ?
fzero is a derivative-free method mainly based on bisection.
How does fsolve differ from fzero ?
fsolve iterations are based on the derivatives of the functions (like Newton's method).
If you need further information, you should look at the pages
under "Algorithms" and "References".
VIGNESH BALAJI
2024-2-6
@Matt J Yes, I will do it. I have a few more quesstions related to this thread. Should I raise them as new questions ?
Matt J
2024-2-6
If your questions are related to what you posted, then it wouldn't make sense to start a new thread. So far, hwoever, none of your follow-up questions appear to be related.
VIGNESH BALAJI
2024-2-6
编辑:VIGNESH BALAJI
2024-2-6
@Dyuman Joshi, @Torsten and @Matt J, based on the iterations of fzero above. At the iteration number 23 it shows the correct answer also with a sign change and why does it still show NaN as the end result even after iterating through the correct answer.
@Dyuman Joshi and @Matt J Both of your suggestions of plotting the function to understand it, helped me a lot. @Torsten Thanks for the suggestion of fzero.
@Matt J After this question, I will close this thread by accepting an answer. For all future not so closely related questions, I will open a new thread.
Torsten
2024-2-6
编辑:Torsten
2024-2-6
At the iteration number 23 it shows the correct answer also with a sign change and why does it still show NaN as the end result even after iterating through the correct answer.
Can you show us what you mean by changing the output of the code below accordingly ?
dd = 0.1069;
vv = 0.7889;
L = 1;
xx0 = 0.15;
aa = fzero(@(x)fun(x,dd,vv,L),xx0)
aa = 0.1500
aa = 0.1458
aa = 0.1542
aa = 0.1440
aa = 0.1560
aa = 0.1415
aa = 0.1585
aa = 0.1380
aa = 0.1620
aa = 0.1330
aa = 0.1670
aa = 0.1260
aa = 0.1740
aa = 0.1161
aa = 0.1839
aa = 0.1020
aa = 0.1980
aa = 0.0821
aa = 0.2179
aa = 0.0540
aa = 0.2460
aa = 0.0142
aa = 0.2858
aa = -0.0420
aa = 0.3420
aa = -0.1215
aa = 0.4215
aa = -0.2340
aa = 0.5340
aa = -0.3931
aa = 0.6931
aa = -0.6180
aa = 0.9180
aa = -0.9361
aa = 1.2361
aa = -1.3860
aa = 1.6860
aa = -2.0222
aa = 2.3222
aa = -2.9220
aa = 3.2220
aa = -4.1945
aa = 4.4945
aa = -5.9940
aa = 6.2940
aa = -8.5389
aa = 8.8389
aa = -12.1380
aa = 12.4380
aa = -17.2279
aa = 17.5279
aa = -24.4260
aa = 24.7260
aa = -34.6057
aa = 34.9057
aa = -49.0020
aa = 49.3020
aa = -69.3614
aa = 69.6614
aa = -98.1540
aa = 98.4540
aa = -138.8729
aa = 139.1729
aa = -196.4580
aa = 196.7580
aa = -277.8957
aa = 278.1957
aa = -393.0660
aa = 393.3660
aa = -555.9414
aa = 556.2414
aa = -786.2820
aa = 786.5820
aa = -1.1120e+03
aa = 1.1123e+03
aa = -1.5727e+03
aa = 1.5730e+03
aa = -2.2242e+03
aa = 2.2245e+03
aa = -3.1456e+03
aa = 3.1459e+03
aa = -4.4486e+03
aa = 4.4489e+03
aa = -6.2913e+03
aa = 6.2916e+03
aa = -8.8973e+03
aa = 8.8976e+03
aa = -1.2583e+04
aa = 1.2583e+04
aa = -1.7795e+04
aa = 1.7795e+04
aa = -2.5166e+04
aa = 2.5166e+04
aa = -3.5590e+04
aa = 3.5590e+04
aa = -5.0331e+04
aa = 5.0332e+04
aa = -7.1180e+04
aa = 7.1180e+04
aa = -1.0066e+05
aa = 1.0066e+05
aa = -1.4236e+05
aa = 1.4236e+05
aa = -2.0133e+05
aa = 2.0133e+05
aa = -2.8472e+05
aa = 2.8472e+05
aa = -4.0265e+05
aa = 4.0265e+05
aa = -5.6944e+05
aa = 5.6944e+05
aa = -8.0531e+05
aa = 8.0531e+05
aa = -1.1389e+06
aa = 1.1389e+06
aa = -1.6106e+06
aa = 1.6106e+06
aa = -2.2778e+06
aa = 2.2778e+06
aa = -3.2212e+06
aa = 3.2212e+06
aa = -4.5555e+06
aa = 4.5555e+06
aa = -6.4425e+06
aa = 6.4425e+06
aa = -9.1110e+06
aa = 9.1110e+06
aa = -1.2885e+07
aa = 1.2885e+07
aa = -1.8222e+07
aa = 1.8222e+07
aa = -2.5770e+07
aa = 2.5770e+07
aa = -3.6444e+07
aa = 3.6444e+07
aa = -5.1540e+07
aa = 5.1540e+07
aa = -7.2888e+07
aa = 7.2888e+07
aa = -1.0308e+08
aa = 1.0308e+08
aa = -1.4578e+08
aa = 1.4578e+08
aa = -2.0616e+08
aa = 2.0616e+08
aa = -2.9155e+08
aa = 2.9155e+08
aa = -4.1232e+08
aa = 4.1232e+08
aa = -5.8310e+08
aa = 5.8310e+08
aa = -8.2463e+08
aa = 8.2463e+08
aa = -1.1662e+09
aa = 1.1662e+09
aa = -1.6493e+09
aa = 1.6493e+09
aa = -2.3324e+09
aa = 2.3324e+09
aa = -3.2985e+09
aa = 3.2985e+09
aa = -4.6648e+09
aa = 4.6648e+09
aa = -6.5971e+09
aa = 6.5971e+09
aa = -9.3297e+09
aa = 9.3297e+09
aa = -1.3194e+10
aa = 1.3194e+10
aa = -1.8659e+10
aa = 1.8659e+10
aa = -2.6388e+10
aa = 2.6388e+10
aa = -3.7319e+10
aa = 3.7319e+10
aa = -5.2777e+10
aa = 5.2777e+10
aa = -7.4637e+10
aa = 7.4637e+10
aa = -1.0555e+11
aa = 1.0555e+11
aa = -1.4927e+11
aa = 1.4927e+11
aa = -2.1111e+11
aa = 2.1111e+11
aa = -2.9855e+11
aa = 2.9855e+11
aa = -4.2221e+11
aa = 4.2221e+11
aa = -5.9710e+11
aa = 5.9710e+11
aa = -8.4442e+11
aa = 8.4442e+11
aa = -1.1942e+12
aa = 1.1942e+12
aa = -1.6888e+12
aa = 1.6888e+12
aa = -2.3884e+12
aa = 2.3884e+12
aa = -3.3777e+12
aa = 3.3777e+12
aa = -4.7768e+12
aa = 4.7768e+12
aa = -6.7554e+12
aa = 6.7554e+12
aa = -9.5536e+12
aa = 9.5536e+12
aa = -1.3511e+13
aa = 1.3511e+13
aa = -1.9107e+13
aa = 1.9107e+13
aa = -2.7022e+13
aa = 2.7022e+13
aa = -3.8214e+13
aa = 3.8214e+13
aa = -5.4043e+13
aa = 5.4043e+13
aa = -7.6429e+13
aa = 7.6429e+13
aa = -1.0809e+14
aa = 1.0809e+14
aa = -1.5286e+14
aa = 1.5286e+14
aa = -2.1617e+14
aa = 2.1617e+14
aa = -3.0571e+14
aa = 3.0571e+14
aa = -4.3235e+14
aa = 4.3235e+14
aa = -6.1143e+14
aa = 6.1143e+14
aa = -8.6469e+14
aa = 8.6469e+14
aa = -1.2229e+15
aa = 1.2229e+15
aa = -1.7294e+15
aa = 1.7294e+15
aa = -2.4457e+15
aa = 2.4457e+15
aa = -3.4588e+15
aa = 3.4588e+15
aa = -4.8914e+15
aa = 4.8914e+15
aa = -6.9175e+15
aa = 6.9175e+15
aa = -9.7829e+15
aa = 9.7829e+15
aa = -1.3835e+16
aa = 1.3835e+16
aa = -1.9566e+16
aa = 1.9566e+16
aa = -2.7670e+16
aa = 2.7670e+16
aa = -3.9131e+16
aa = 3.9131e+16
aa = -5.5340e+16
aa = 5.5340e+16
aa = -7.8263e+16
aa = 7.8263e+16
aa = -1.1068e+17
aa = 1.1068e+17
aa = -1.5653e+17
aa = 1.5653e+17
aa = -2.2136e+17
aa = 2.2136e+17
aa = -3.1305e+17
aa = 3.1305e+17
aa = -4.4272e+17
aa = 4.4272e+17
aa = -6.2610e+17
aa = 6.2610e+17
aa = -8.8544e+17
aa = 8.8544e+17
aa = -1.2522e+18
aa = 1.2522e+18
aa = -1.7709e+18
aa = 1.7709e+18
aa = -2.5044e+18
aa = 2.5044e+18
aa = -3.5418e+18
aa = 3.5418e+18
aa = -5.0088e+18
aa = 5.0088e+18
aa = -7.0835e+18
aa = 7.0835e+18
aa = -1.0018e+19
aa = 1.0018e+19
aa = -1.4167e+19
aa = 1.4167e+19
aa = -2.0035e+19
aa = 2.0035e+19
aa = -2.8334e+19
aa = 2.8334e+19
aa = -4.0071e+19
aa = 4.0071e+19
aa = -5.6668e+19
aa = 5.6668e+19
aa = -8.0141e+19
aa = 8.0141e+19
aa = -1.1334e+20
aa = 1.1334e+20
aa = -1.6028e+20
aa = 1.6028e+20
aa = -2.2667e+20
aa = 2.2667e+20
aa = -3.2056e+20
aa = 3.2056e+20
aa = -4.5335e+20
aa = 4.5335e+20
aa = -6.4113e+20
aa = 6.4113e+20
aa = -9.0669e+20
aa = 9.0669e+20
aa = -1.2823e+21
aa = 1.2823e+21
aa = -1.8134e+21
aa = 1.8134e+21
aa = -2.5645e+21
aa = 2.5645e+21
aa = -3.6268e+21
aa = 3.6268e+21
aa = -5.1290e+21
aa = 5.1290e+21
aa = -7.2536e+21
aa = 7.2536e+21
aa = -1.0258e+22
aa = 1.0258e+22
aa = -1.4507e+22
aa = 1.4507e+22
aa = -2.0516e+22
aa = 2.0516e+22
aa = -2.9014e+22
aa = 2.9014e+22
aa = -4.1032e+22
aa = 4.1032e+22
aa = -5.8028e+22
aa = 5.8028e+22
aa = -8.2065e+22
aa = 8.2065e+22
aa = -1.1606e+23
aa = 1.1606e+23
aa = -1.6413e+23
aa = 1.6413e+23
aa = -2.3211e+23
aa = 2.3211e+23
aa = -3.2826e+23
aa = 3.2826e+23
aa = -4.6423e+23
aa = 4.6423e+23
aa = -6.5652e+23
aa = 6.5652e+23
aa = -9.2846e+23
aa = 9.2846e+23
aa = -1.3130e+24
aa = 1.3130e+24
aa = -1.8569e+24
aa = 1.8569e+24
aa = -2.6261e+24
aa = 2.6261e+24
aa = -3.7138e+24
aa = 3.7138e+24
aa = -5.2521e+24
aa = 5.2521e+24
aa = -7.4276e+24
aa = 7.4276e+24
aa = -1.0504e+25
aa = 1.0504e+25
aa = -1.4855e+25
aa = 1.4855e+25
aa = -2.1009e+25
aa = 2.1009e+25
aa = -2.9711e+25
aa = 2.9711e+25
aa = -4.2017e+25
aa = 4.2017e+25
aa = -5.9421e+25
aa = 5.9421e+25
aa = -8.4034e+25
aa = 8.4034e+25
aa = -1.1884e+26
aa = 1.1884e+26
aa = -1.6807e+26
aa = 1.6807e+26
aa = -2.3768e+26
aa = 2.3768e+26
aa = -3.3614e+26
aa = 3.3614e+26
aa = -4.7537e+26
aa = 4.7537e+26
aa = -6.7227e+26
aa = 6.7227e+26
aa = -9.5074e+26
aa = 9.5074e+26
aa = -1.3445e+27
aa = 1.3445e+27
aa = -1.9015e+27
aa = 1.9015e+27
aa = -2.6891e+27
aa = 2.6891e+27
aa = -3.8030e+27
aa = 3.8030e+27
aa = -5.3782e+27
aa = 5.3782e+27
aa = -7.6059e+27
aa = 7.6059e+27
aa = -1.0756e+28
aa = 1.0756e+28
aa = -1.5212e+28
aa = 1.5212e+28
aa = -2.1513e+28
aa = 2.1513e+28
aa = -3.0424e+28
aa = 3.0424e+28
aa = -4.3025e+28
aa = 4.3025e+28
aa = -6.0847e+28
aa = 6.0847e+28
aa = -8.6051e+28
aa = 8.6051e+28
aa = -1.2169e+29
aa = 1.2169e+29
aa = -1.7210e+29
aa = 1.7210e+29
aa = -2.4339e+29
aa = 2.4339e+29
aa = -3.4420e+29
aa = 3.4420e+29
aa = -4.8678e+29
aa = 4.8678e+29
aa = -6.8841e+29
aa = 6.8841e+29
aa = -9.7356e+29
aa = 9.7356e+29
aa = -1.3768e+30
aa = 1.3768e+30
aa = -1.9471e+30
aa = 1.9471e+30
aa = -2.7536e+30
aa = 2.7536e+30
aa = -3.8942e+30
aa = 3.8942e+30
aa = -5.5073e+30
aa = 5.5073e+30
aa = -7.7884e+30
aa = 7.7884e+30
aa = -1.1015e+31
aa = 1.1015e+31
aa = -1.5577e+31
aa = 1.5577e+31
aa = -2.2029e+31
aa = 2.2029e+31
aa = -3.1154e+31
aa = 3.1154e+31
aa = -4.4058e+31
aa = 4.4058e+31
aa = -6.2308e+31
aa = 6.2308e+31
aa = -8.8116e+31
aa = 8.8116e+31
aa = -1.2462e+32
aa = 1.2462e+32
aa = -1.7623e+32
aa = 1.7623e+32
aa = -2.4923e+32
aa = 2.4923e+32
aa = -3.5246e+32
aa = 3.5246e+32
aa = -4.9846e+32
aa = 4.9846e+32
aa = -7.0493e+32
aa = 7.0493e+32
aa = -9.9692e+32
aa = 9.9692e+32
aa = -1.4099e+33
aa = 1.4099e+33
aa = -1.9938e+33
aa = 1.9938e+33
aa = -2.8197e+33
aa = 2.8197e+33
aa = -3.9877e+33
aa = 3.9877e+33
aa = -5.6394e+33
aa = 5.6394e+33
aa = -7.9754e+33
aa = 7.9754e+33
aa = -1.1279e+34
aa = 1.1279e+34
aa = -1.5951e+34
aa = 1.5951e+34
aa = -2.2558e+34
aa = 2.2558e+34
aa = -3.1901e+34
aa = 3.1901e+34
aa = -4.5115e+34
aa = 4.5115e+34
aa = -6.3803e+34
aa = 6.3803e+34
aa = -9.0231e+34
aa = 9.0231e+34
aa = -1.2761e+35
aa = 1.2761e+35
aa = -1.8046e+35
aa = 1.8046e+35
aa = -2.5521e+35
aa = 2.5521e+35
aa = -3.6092e+35
aa = 3.6092e+35
aa = -5.1042e+35
aa = 5.1042e+35
aa = -7.2185e+35
aa = 7.2185e+35
aa = -1.0208e+36
aa = 1.0208e+36
aa = -1.4437e+36
aa = 1.4437e+36
aa = -2.0417e+36
aa = 2.0417e+36
aa = -2.8874e+36
aa = 2.8874e+36
aa = -4.0834e+36
aa = 4.0834e+36
aa = -5.7748e+36
aa = 5.7748e+36
aa = -8.1668e+36
aa = 8.1668e+36
aa = -1.1550e+37
aa = 1.1550e+37
aa = -1.6334e+37
aa = 1.6334e+37
aa = -2.3099e+37
aa = 2.3099e+37
aa = -3.2667e+37
aa = 3.2667e+37
aa = -4.6198e+37
aa = 4.6198e+37
aa = -6.5334e+37
aa = 6.5334e+37
aa = -9.2397e+37
aa = 9.2397e+37
aa = -1.3067e+38
aa = 1.3067e+38
aa = -1.8479e+38
aa = 1.8479e+38
aa = -2.6134e+38
aa = 2.6134e+38
aa = -3.6959e+38
aa = 3.6959e+38
aa = -5.2267e+38
aa = 5.2267e+38
aa = -7.3917e+38
aa = 7.3917e+38
aa = -1.0453e+39
aa = 1.0453e+39
aa = -1.4783e+39
aa = 1.4783e+39
aa = -2.0907e+39
aa = 2.0907e+39
aa = -2.9567e+39
aa = 2.9567e+39
aa = -4.1814e+39
aa = 4.1814e+39
aa = -5.9134e+39
aa = 5.9134e+39
aa = -8.3628e+39
aa = 8.3628e+39
aa = -1.1827e+40
aa = 1.1827e+40
aa = -1.6726e+40
aa = 1.6726e+40
aa = -2.3654e+40
aa = 2.3654e+40
aa = -3.3451e+40
aa = 3.3451e+40
aa = -4.7307e+40
aa = 4.7307e+40
aa = -6.6902e+40
aa = 6.6902e+40
aa = -9.4614e+40
aa = 9.4614e+40
aa = -1.3380e+41
aa = 1.3380e+41
aa = -1.8923e+41
aa = 1.8923e+41
aa = -2.6761e+41
aa = 2.6761e+41
aa = -3.7846e+41
aa = 3.7846e+41
aa = -5.3522e+41
aa = 5.3522e+41
aa = -7.5691e+41
aa = 7.5691e+41
aa = -1.0704e+42
aa = 1.0704e+42
aa = -1.5138e+42
aa = 1.5138e+42
aa = -2.1409e+42
aa = 2.1409e+42
aa = -3.0276e+42
aa = 3.0276e+42
aa = -4.2817e+42
aa = 4.2817e+42
aa = -6.0553e+42
aa = 6.0553e+42
aa = -8.5635e+42
aa = 8.5635e+42
aa = -1.2111e+43
aa = 1.2111e+43
aa = -1.7127e+43
aa = 1.7127e+43
aa = -2.4221e+43
aa = 2.4221e+43
aa = -3.4254e+43
aa = 3.4254e+43
aa = -4.8442e+43
aa = 4.8442e+43
aa = -6.8508e+43
aa = 6.8508e+43
aa = -9.6885e+43
aa = 9.6885e+43
aa = -1.3702e+44
aa = 1.3702e+44
aa = -1.9377e+44
aa = 1.9377e+44
aa = -2.7403e+44
aa = 2.7403e+44
aa = -3.8754e+44
aa = 3.8754e+44
aa = -5.4806e+44
aa = 5.4806e+44
aa = -7.7508e+44
aa = 7.7508e+44
aa = -1.0961e+45
aa = 1.0961e+45
aa = -1.5502e+45
aa = 1.5502e+45
aa = -2.1923e+45
aa = 2.1923e+45
aa = -3.1003e+45
aa = 3.1003e+45
aa = -4.3845e+45
aa = 4.3845e+45
aa = -6.2006e+45
aa = 6.2006e+45
aa = -8.7690e+45
aa = 8.7690e+45
aa = -1.2401e+46
aa = 1.2401e+46
aa = -1.7538e+46
aa = 1.7538e+46
aa = -2.4803e+46
aa = 2.4803e+46
aa = -3.5076e+46
aa = 3.5076e+46
aa = -4.9605e+46
aa = 4.9605e+46
aa = -7.0152e+46
aa = 7.0152e+46
aa = -9.9210e+46
aa = 9.9210e+46
aa = -1.4030e+47
aa = 1.4030e+47
aa = -1.9842e+47
aa = 1.9842e+47
aa = -2.8061e+47
aa = 2.8061e+47
aa = -3.9684e+47
aa = 3.9684e+47
aa = -5.6122e+47
aa = 5.6122e+47
aa = -7.9368e+47
aa = 7.9368e+47
aa = -1.1224e+48
aa = 1.1224e+48
aa = -1.5874e+48
aa = 1.5874e+48
aa = -2.2449e+48
aa = 2.2449e+48
aa = -3.1747e+48
aa = 3.1747e+48
aa = -4.4897e+48
aa = 4.4897e+48
aa = -6.3494e+48
aa = 6.3494e+48
aa = -8.9795e+48
aa = 8.9795e+48
aa = -1.2699e+49
aa = 1.2699e+49
aa = -1.7959e+49
aa = 1.7959e+49
aa = -2.5398e+49
aa = 2.5398e+49
aa = -3.5918e+49
aa = 3.5918e+49
aa = -5.0796e+49
aa = 5.0796e+49
aa = -7.1836e+49
aa = 7.1836e+49
aa = -1.0159e+50
aa = 1.0159e+50
aa = -1.4367e+50
aa = 1.4367e+50
aa = -2.0318e+50
aa = 2.0318e+50
aa = -2.8734e+50
aa = 2.8734e+50
aa = -4.0636e+50
aa = 4.0636e+50
aa = -5.7469e+50
aa = 5.7469e+50
aa = -8.1273e+50
aa = 8.1273e+50
aa = -1.1494e+51
aa = 1.1494e+51
aa = -1.6255e+51
aa = 1.6255e+51
aa = -2.2987e+51
aa = 2.2987e+51
aa = -3.2509e+51
aa = 3.2509e+51
aa = -4.5975e+51
aa = 4.5975e+51
aa = -6.5018e+51
aa = 6.5018e+51
aa = -9.1950e+51
aa = 9.1950e+51
aa = -1.3004e+52
aa = 1.3004e+52
aa = -1.8390e+52
aa = 1.8390e+52
aa = -2.6007e+52
aa = 2.6007e+52
aa = -3.6780e+52
aa = 3.6780e+52
aa = -5.2015e+52
aa = 5.2015e+52
aa = -7.3560e+52
aa = 7.3560e+52
aa = -1.0403e+53
aa = 1.0403e+53
aa = -1.4712e+53
aa = 1.4712e+53
aa = -2.0806e+53
aa = 2.0806e+53
aa = -2.9424e+53
aa = 2.9424e+53
aa = -4.1612e+53
aa = 4.1612e+53
aa = -5.8848e+53
aa = 5.8848e+53
aa = -8.3223e+53
aa = 8.3223e+53
aa = -1.1770e+54
aa = 1.1770e+54
aa = -1.6645e+54
aa = 1.6645e+54
aa = -2.3539e+54
aa = 2.3539e+54
aa = -3.3289e+54
aa = 3.3289e+54
aa = -4.7078e+54
aa = 4.7078e+54
aa = -6.6579e+54
aa = 6.6579e+54
aa = -9.4157e+54
aa = 9.4157e+54
aa = -1.3316e+55
aa = 1.3316e+55
aa = -1.8831e+55
aa = 1.8831e+55
aa = -2.6631e+55
aa = 2.6631e+55
aa = -3.7663e+55
aa = 3.7663e+55
aa = -5.3263e+55
aa = 5.3263e+55
aa = -7.5325e+55
aa = 7.5325e+55
aa = -1.0653e+56
aa = 1.0653e+56
aa = -1.5065e+56
aa = 1.5065e+56
aa = -2.1305e+56
aa = 2.1305e+56
aa = -3.0130e+56
aa = 3.0130e+56
aa = -4.2610e+56
aa = 4.2610e+56
aa = -6.0260e+56
aa = 6.0260e+56
aa = -8.5221e+56
aa = 8.5221e+56
aa = -1.2052e+57
aa = 1.2052e+57
aa = -1.7044e+57
aa = 1.7044e+57
aa = -2.4104e+57
aa = 2.4104e+57
aa = -3.4088e+57
aa = 3.4088e+57
aa = -4.8208e+57
aa = 4.8208e+57
aa = -6.8177e+57
aa = 6.8177e+57
aa = -9.6416e+57
aa = 9.6416e+57
aa = -1.3635e+58
aa = 1.3635e+58
aa = -1.9283e+58
aa = 1.9283e+58
aa = -2.7271e+58
aa = 2.7271e+58
aa = -3.8567e+58
aa = 3.8567e+58
aa = -5.4541e+58
aa = 5.4541e+58
aa = -7.7133e+58
aa = 7.7133e+58
aa = -1.0908e+59
aa = 1.0908e+59
aa = -1.5427e+59
aa = 1.5427e+59
aa = -2.1817e+59
aa = 2.1817e+59
aa = -3.0853e+59
aa = 3.0853e+59
aa = -4.3633e+59
aa = 4.3633e+59
aa = -6.1706e+59
aa = 6.1706e+59
aa = -8.7266e+59
aa = 8.7266e+59
aa = -1.2341e+60
aa = 1.2341e+60
aa = -1.7453e+60
aa = 1.7453e+60
aa = -2.4683e+60
aa = 2.4683e+60
aa = -3.4906e+60
aa = 3.4906e+60
aa = -4.9365e+60
aa = 4.9365e+60
aa = -6.9813e+60
aa = 6.9813e+60
aa = -9.8730e+60
aa = 9.8730e+60
aa = -1.3963e+61
aa = 1.3963e+61
aa = -1.9746e+61
aa = 1.9746e+61
aa = -2.7925e+61
aa = 2.7925e+61
aa = -3.9492e+61
aa = 3.9492e+61
aa = -5.5850e+61
aa = 5.5850e+61
aa = -7.8984e+61
aa = 7.8984e+61
aa = -1.1170e+62
aa = 1.1170e+62
aa = -1.5797e+62
aa = 1.5797e+62
aa = -2.2340e+62
aa = 2.2340e+62
aa = -3.1594e+62
aa = 3.1594e+62
aa = -4.4680e+62
aa = 4.4680e+62
aa = -6.3187e+62
aa = 6.3187e+62
aa = -8.9360e+62
aa = 8.9360e+62
aa = -1.2637e+63
aa = 1.2637e+63
aa = -1.7872e+63
aa = 1.7872e+63
aa = -2.5275e+63
aa = 2.5275e+63
aa = -3.5744e+63
aa = 3.5744e+63
aa = -5.0550e+63
aa = 5.0550e+63
aa = -7.1488e+63
aa = 7.1488e+63
aa = -1.0110e+64
aa = 1.0110e+64
aa = -1.4298e+64
aa = 1.4298e+64
aa = -2.0220e+64
aa = 2.0220e+64
aa = -2.8595e+64
aa = 2.8595e+64
aa = -4.0440e+64
aa = 4.0440e+64
aa = -5.7191e+64
aa = 5.7191e+64
aa = -8.0880e+64
aa = 8.0880e+64
aa = -1.1438e+65
aa = 1.1438e+65
aa = -1.6176e+65
aa = 1.6176e+65
aa = -2.2876e+65
aa = 2.2876e+65
aa = -3.2352e+65
aa = 3.2352e+65
aa = -4.5753e+65
aa = 4.5753e+65
aa = -6.4704e+65
aa = 6.4704e+65
aa = -9.1505e+65
aa = 9.1505e+65
aa = -1.2941e+66
aa = 1.2941e+66
aa = -1.8301e+66
aa = 1.8301e+66
aa = -2.5882e+66
aa = 2.5882e+66
aa = -3.6602e+66
aa = 3.6602e+66
aa = -5.1763e+66
aa = 5.1763e+66
aa = -7.3204e+66
aa = 7.3204e+66
aa = -1.0353e+67
aa = 1.0353e+67
aa = -1.4641e+67
aa = 1.4641e+67
aa = -2.0705e+67
aa = 2.0705e+67
aa = -2.9282e+67
aa = 2.9282e+67
aa = -4.1410e+67
aa = 4.1410e+67
aa = -5.8563e+67
aa = 5.8563e+67
aa = -8.2821e+67
aa = 8.2821e+67
aa = -1.1713e+68
aa = 1.1713e+68
aa = -1.6564e+68
aa = 1.6564e+68
aa = -2.3425e+68
aa = 2.3425e+68
aa = -3.3128e+68
aa = 3.3128e+68
aa = -4.6851e+68
aa = 4.6851e+68
aa = -6.6257e+68
aa = 6.6257e+68
aa = -9.3701e+68
aa = 9.3701e+68
aa = -1.3251e+69
aa = 1.3251e+69
aa = -1.8740e+69
aa = 1.8740e+69
aa = -2.6503e+69
aa = 2.6503e+69
aa = -3.7480e+69
aa = 3.7480e+69
aa = -5.3005e+69
aa = 5.3005e+69
aa = -7.4961e+69
aa = 7.4961e+69
aa = -1.0601e+70
aa = 1.0601e+70
aa = -1.4992e+70
aa = 1.4992e+70
aa = -2.1202e+70
aa = 2.1202e+70
aa = -2.9984e+70
aa = 2.9984e+70
aa = -4.2404e+70
aa = 4.2404e+70
aa = -5.9969e+70
aa = 5.9969e+70
aa = -8.4809e+70
aa = 8.4809e+70
aa = -1.1994e+71
aa = 1.1994e+71
aa = -1.6962e+71
aa = 1.6962e+71
aa = -2.3988e+71
aa = 2.3988e+71
aa = -3.3923e+71
aa = 3.3923e+71
aa = -4.7975e+71
aa = 4.7975e+71
aa = -6.7847e+71
aa = 6.7847e+71
aa = -9.5950e+71
aa = 9.5950e+71
aa = -1.3569e+72
aa = 1.3569e+72
aa = -1.9190e+72
aa = 1.9190e+72
aa = -2.7139e+72
aa = 2.7139e+72
aa = -3.8380e+72
aa = 3.8380e+72
aa = -5.4278e+72
aa = 5.4278e+72
aa = -7.6760e+72
aa = 7.6760e+72
aa = -1.0856e+73
aa = 1.0856e+73
aa = -1.5352e+73
aa = 1.5352e+73
aa = -2.1711e+73
aa = 2.1711e+73
aa = -3.0704e+73
aa = 3.0704e+73
aa = -4.3422e+73
aa = 4.3422e+73
aa = -6.1408e+73
aa = 6.1408e+73
aa = -8.6844e+73
aa = 8.6844e+73
aa = -1.2282e+74
aa = 1.2282e+74
aa = -1.7369e+74
aa = 1.7369e+74
aa = -2.4563e+74
aa = 2.4563e+74
aa = -3.4738e+74
aa = 3.4738e+74
aa = -4.9126e+74
aa = 4.9126e+74
aa = -6.9475e+74
aa = 6.9475e+74
aa = -9.8253e+74
aa = 9.8253e+74
aa = -1.3895e+75
aa = 1.3895e+75
aa = -1.9651e+75
aa = 1.9651e+75
aa = -2.7790e+75
aa = 2.7790e+75
aa = -3.9301e+75
aa = 3.9301e+75
aa = -5.5580e+75
aa = 5.5580e+75
aa = -7.8602e+75
aa = 7.8602e+75
aa = -1.1116e+76
aa = 1.1116e+76
aa = -1.5720e+76
aa = 1.5720e+76
aa = -2.2232e+76
aa = 2.2232e+76
aa = -3.1441e+76
aa = 3.1441e+76
aa = -4.4464e+76
aa = 4.4464e+76
aa = -6.2882e+76
aa = 6.2882e+76
aa = -8.8928e+76
aa = 8.8928e+76
aa = -1.2576e+77
aa = 1.2576e+77
aa = -1.7786e+77
aa = 1.7786e+77
aa = -2.5153e+77
aa = 2.5153e+77
aa = -3.5571e+77
aa = 3.5571e+77
aa = -5.0305e+77
aa = 5.0305e+77
aa = -7.1143e+77
aa = 7.1143e+77
aa = -1.0061e+78
aa = 1.0061e+78
aa = -1.4229e+78
aa = 1.4229e+78
aa = -2.0122e+78
aa = 2.0122e+78
aa = -2.8457e+78
aa = 2.8457e+78
aa = -4.0244e+78
aa = 4.0244e+78
aa = -5.6914e+78
aa = 5.6914e+78
aa = -8.0489e+78
aa = 8.0489e+78
aa = -1.1383e+79
aa = 1.1383e+79
aa = -1.6098e+79
aa = 1.6098e+79
aa = -2.2766e+79
aa = 2.2766e+79
aa = -3.2195e+79
aa = 3.2195e+79
aa = -4.5531e+79
aa = 4.5531e+79
aa = -6.4391e+79
aa = 6.4391e+79
aa = -9.1063e+79
aa = 9.1063e+79
aa = -1.2878e+80
aa = 1.2878e+80
aa = -1.8213e+80
aa = 1.8213e+80
aa = -2.5756e+80
aa = 2.5756e+80
aa = -3.6425e+80
aa = 3.6425e+80
aa = -5.1513e+80
aa = 5.1513e+80
aa = -7.2850e+80
aa = 7.2850e+80
aa = -1.0303e+81
aa = 1.0303e+81
aa = -1.4570e+81
aa = 1.4570e+81
aa = -2.0605e+81
aa = 2.0605e+81
aa = -2.9140e+81
aa = 2.9140e+81
aa = -4.1210e+81
aa = 4.1210e+81
aa = -5.8280e+81
aa = 5.8280e+81
aa = -8.2420e+81
aa = 8.2420e+81
aa = -1.1656e+82
aa = 1.1656e+82
aa = -1.6484e+82
aa = 1.6484e+82
aa = -2.3312e+82
aa = 2.3312e+82
aa = -3.2968e+82
aa = 3.2968e+82
aa = -4.6624e+82
aa = 4.6624e+82
aa = -6.5936e+82
aa = 6.5936e+82
aa = -9.3248e+82
aa = 9.3248e+82
aa = -1.3187e+83
aa = 1.3187e+83
aa = -1.8650e+83
aa = 1.8650e+83
aa = -2.6375e+83
aa = 2.6375e+83
aa = -3.7299e+83
aa = 3.7299e+83
aa = -5.2749e+83
aa = 5.2749e+83
aa = -7.4598e+83
aa = 7.4598e+83
aa = -1.0550e+84
aa = 1.0550e+84
aa = -1.4920e+84
aa = 1.4920e+84
aa = -2.1100e+84
aa = 2.1100e+84
aa = -2.9839e+84
aa = 2.9839e+84
aa = -4.2199e+84
aa = 4.2199e+84
aa = -5.9679e+84
aa = 5.9679e+84
aa = -8.4399e+84
aa = 8.4399e+84
aa = -1.1936e+85
aa = 1.1936e+85
aa = -1.6880e+85
aa = 1.6880e+85
aa = -2.3872e+85
aa = 2.3872e+85
aa = -3.3759e+85
aa = 3.3759e+85
aa = -4.7743e+85
aa = 4.7743e+85
aa = -6.7519e+85
aa = 6.7519e+85
aa = -9.5486e+85
aa = 9.5486e+85
aa = -1.3504e+86
aa = 1.3504e+86
aa = -1.9097e+86
aa = 1.9097e+86
aa = -2.7008e+86
aa = 2.7008e+86
aa = -3.8194e+86
aa = 3.8194e+86
aa = -5.4015e+86
aa = 5.4015e+86
aa = -7.6389e+86
aa = 7.6389e+86
aa = -1.0803e+87
aa = 1.0803e+87
aa = -1.5278e+87
aa = 1.5278e+87
aa = -2.1606e+87
aa = 2.1606e+87
aa = -3.0556e+87
aa = 3.0556e+87
aa = -4.3212e+87
aa = 4.3212e+87
aa = -6.1111e+87
aa = 6.1111e+87
aa = -8.6424e+87
aa = 8.6424e+87
aa = -1.2222e+88
aa = 1.2222e+88
aa = -1.7285e+88
aa = 1.7285e+88
aa = -2.4444e+88
aa = 2.4444e+88
aa = -3.4570e+88
aa = 3.4570e+88
aa = -4.8889e+88
aa = 4.8889e+88
aa = -6.9139e+88
aa = 6.9139e+88
aa = -9.7778e+88
aa = 9.7778e+88
aa = -1.3828e+89
aa = 1.3828e+89
aa = -1.9556e+89
aa = 1.9556e+89
aa = -2.7656e+89
aa = 2.7656e+89
aa = -3.9111e+89
aa = 3.9111e+89
aa = -5.5311e+89
aa = 5.5311e+89
aa = -7.8222e+89
aa = 7.8222e+89
aa = -1.1062e+90
aa = 1.1062e+90
aa = -1.5644e+90
aa = 1.5644e+90
aa = -2.2125e+90
aa = 2.2125e+90
aa = -3.1289e+90
aa = 3.1289e+90
aa = -4.4249e+90
aa = 4.4249e+90
aa = -6.2578e+90
aa = 6.2578e+90
aa = -8.8498e+90
aa = 8.8498e+90
aa = -1.2516e+91
aa = 1.2516e+91
aa = -1.7700e+91
aa = 1.7700e+91
aa = -2.5031e+91
aa = 2.5031e+91
aa = -3.5399e+91
aa = 3.5399e+91
aa = -5.0062e+91
aa = 5.0062e+91
aa = -7.0799e+91
aa = 7.0799e+91
aa = -1.0012e+92
aa = 1.0012e+92
aa = -1.4160e+92
aa = 1.4160e+92
aa = -2.0025e+92
aa = 2.0025e+92
aa = -2.8319e+92
aa = 2.8319e+92
aa = -4.0050e+92
aa = 4.0050e+92
aa = -5.6639e+92
aa = 5.6639e+92
aa = -8.0100e+92
aa = 8.0100e+92
aa = -1.1328e+93
aa = 1.1328e+93
aa = -1.6020e+93
aa = 1.6020e+93
aa = -2.2656e+93
aa = 2.2656e+93
aa = -3.2040e+93
aa = 3.2040e+93
aa = -4.5311e+93
aa = 4.5311e+93
aa = -6.4080e+93
aa = 6.4080e+93
aa = -9.0622e+93
aa = 9.0622e+93
aa = -1.2816e+94
aa = 1.2816e+94
aa = -1.8124e+94
aa = 1.8124e+94
aa = -2.5632e+94
aa = 2.5632e+94
aa = -3.6249e+94
aa = 3.6249e+94
aa = -5.1264e+94
aa = 5.1264e+94
aa = -7.2498e+94
aa = 7.2498e+94
aa = -1.0253e+95
aa = 1.0253e+95
aa = -1.4500e+95
aa = 1.4500e+95
aa = -2.0505e+95
aa = 2.0505e+95
aa = -2.8999e+95
aa = 2.8999e+95
aa = -4.1011e+95
aa = 4.1011e+95
aa = -5.7998e+95
aa = 5.7998e+95
aa = -8.2022e+95
aa = 8.2022e+95
aa = -1.1600e+96
aa = 1.1600e+96
aa = -1.6404e+96
aa = 1.6404e+96
aa = -2.3199e+96
aa = 2.3199e+96
aa = -3.2809e+96
aa = 3.2809e+96
aa = -4.6399e+96
aa = 4.6399e+96
aa = -6.5618e+96
aa = 6.5618e+96
aa = -9.2797e+96
aa = 9.2797e+96
aa = -1.3124e+97
aa = 1.3124e+97
aa = -1.8559e+97
aa = 1.8559e+97
aa = -2.6247e+97
aa = 2.6247e+97
aa = -3.7119e+97
aa = 3.7119e+97
aa = -5.2494e+97
aa = 5.2494e+97
aa = -7.4238e+97
aa = 7.4238e+97
aa = -1.0499e+98
aa = 1.0499e+98
aa = -1.4848e+98
aa = 1.4848e+98
aa = -2.0998e+98
aa = 2.0998e+98
aa = -2.9695e+98
aa = 2.9695e+98
aa = -4.1995e+98
aa = 4.1995e+98
aa = -5.9390e+98
aa = 5.9390e+98
aa = -8.3990e+98
aa = 8.3990e+98
aa = -1.1878e+99
aa = 1.1878e+99
aa = -1.6798e+99
aa = 1.6798e+99
aa = -2.3756e+99
aa = 2.3756e+99
aa = -3.3596e+99
aa = 3.3596e+99
aa = -4.7512e+99
aa = 4.7512e+99
aa = -6.7192e+99
aa = 6.7192e+99
aa = -9.5024e+99
aa = 9.5024e+99
aa = -1.3438e+100
aa = 1.3438e+100
aa = -1.9005e+100
aa = 1.9005e+100
aa = -2.6877e+100
aa = 2.6877e+100
aa = -3.8010e+100
aa = 3.8010e+100
aa = -5.3754e+100
aa = 5.3754e+100
aa = -7.6019e+100
aa = 7.6019e+100
aa = -1.0751e+101
aa = 1.0751e+101
aa = -1.5204e+101
aa = 1.5204e+101
aa = -2.1502e+101
aa = 2.1502e+101
aa = -3.0408e+101
aa = 3.0408e+101
aa = -4.3003e+101
aa = 4.3003e+101
aa = -6.0816e+101
aa = 6.0816e+101
aa = -8.6006e+101
aa = 8.6006e+101
aa = -1.2163e+102
aa = 1.2163e+102
aa = -1.7201e+102
aa = 1.7201e+102
aa = -2.4326e+102
aa = 2.4326e+102
aa = -3.4402e+102
aa = 3.4402e+102
aa = -4.8652e+102
aa = 4.8652e+102
aa = -6.8805e+102
aa = 6.8805e+102
aa = -9.7305e+102
aa = 9.7305e+102
aa = -1.3761e+103
aa = 1.3761e+103
aa = -1.9461e+103
aa = 1.9461e+103
aa = -2.7522e+103
aa = 2.7522e+103
aa = -3.8922e+103
aa = 3.8922e+103
aa = -5.5044e+103
aa = 5.5044e+103
aa = -7.7844e+103
aa = 7.7844e+103
aa = -1.1009e+104
aa = 1.1009e+104
aa = -1.5569e+104
aa = 1.5569e+104
aa = -2.2018e+104
aa = 2.2018e+104
aa = -3.1138e+104
aa = 3.1138e+104
aa = -4.4035e+104
aa = 4.4035e+104
aa = -6.2275e+104
aa = 6.2275e+104
aa = -8.8070e+104
aa = 8.8070e+104
aa = -1.2455e+105
aa = 1.2455e+105
aa = -1.7614e+105
aa = 1.7614e+105
aa = -2.4910e+105
aa = 2.4910e+105
aa = -3.5228e+105
aa = 3.5228e+105
aa = -4.9820e+105
aa = 4.9820e+105
aa = -7.0456e+105
aa = 7.0456e+105
aa = -9.9640e+105
aa = 9.9640e+105
aa = -1.4091e+106
aa = 1.4091e+106
aa = -1.9928e+106
aa = 1.9928e+106
aa = -2.8183e+106
aa = 2.8183e+106
aa = -3.9856e+106
aa = 3.9856e+106
aa = -5.6365e+106
aa = 5.6365e+106
aa = -7.9712e+106
aa = 7.9712e+106
aa = -1.1273e+107
aa = 1.1273e+107
aa = -1.5942e+107
aa = 1.5942e+107
aa = -2.2546e+107
aa = 2.2546e+107
aa = -3.1885e+107
aa = 3.1885e+107
aa = -4.5092e+107
aa = 4.5092e+107
aa = -6.3770e+107
aa = 6.3770e+107
aa = -9.0184e+107
aa = 9.0184e+107
aa = -1.2754e+108
aa = 1.2754e+108
aa = -1.8037e+108
aa = 1.8037e+108
aa = -2.5508e+108
aa = 2.5508e+108
aa = -3.6074e+108
aa = 3.6074e+108
aa = -5.1016e+108
aa = 5.1016e+108
aa = -7.2147e+108
aa = 7.2147e+108
aa = -1.0203e+109
aa = 1.0203e+109
aa = -1.4429e+109
aa = 1.4429e+109
aa = -2.0406e+109
aa = 2.0406e+109
aa = -2.8859e+109
aa = 2.8859e+109
aa = -4.0813e+109
aa = 4.0813e+109
aa = -5.7718e+109
aa = 5.7718e+109
aa = -8.1625e+109
aa = 8.1625e+109
aa = -1.1544e+110
aa = 1.1544e+110
aa = -1.6325e+110
aa = 1.6325e+110
aa = -2.3087e+110
aa = 2.3087e+110
aa = -3.2650e+110
aa = 3.2650e+110
aa = -4.6174e+110
aa = 4.6174e+110
aa = -6.5300e+110
aa = 6.5300e+110
aa = -9.2348e+110
aa = 9.2348e+110
aa = -1.3060e+111
aa = 1.3060e+111
aa = -1.8470e+111
aa = 1.8470e+111
aa = -2.6120e+111
aa = 2.6120e+111
aa = -3.6939e+111
aa = 3.6939e+111
aa = -5.2240e+111
aa = 5.2240e+111
aa = -7.3879e+111
aa = 7.3879e+111
aa = -1.0448e+112
aa = 1.0448e+112
aa = -1.4776e+112
aa = 1.4776e+112
aa = -2.0896e+112
aa = 2.0896e+112
aa = -2.9552e+112
aa = 2.9552e+112
aa = -4.1792e+112
aa = 4.1792e+112
aa = -5.9103e+112
aa = 5.9103e+112
aa = -8.3584e+112
aa = 8.3584e+112
aa = -1.1821e+113
aa = 1.1821e+113
aa = -1.6717e+113
aa = 1.6717e+113
aa = -2.3641e+113
aa = 2.3641e+113
aa = -3.3434e+113
aa = 3.3434e+113
aa = -4.7282e+113
aa = 4.7282e+113
aa = -6.6867e+113
aa = 6.6867e+113
aa = -9.4565e+113
aa = 9.4565e+113
aa = -1.3373e+114
aa = 1.3373e+114
aa = -1.8913e+114
aa = 1.8913e+114
aa = -2.6747e+114
aa = 2.6747e+114
aa = -3.7826e+114
aa = 3.7826e+114
aa = -5.3494e+114
aa = 5.3494e+114
aa = -7.5652e+114
aa = 7.5652e+114
aa = -1.0699e+115
aa = 1.0699e+115
aa = -1.5130e+115
aa = 1.5130e+115
aa = -2.1398e+115
aa = 2.1398e+115
aa = -3.0261e+115
aa = 3.0261e+115
aa = -4.2795e+115
aa = 4.2795e+115
aa = -6.0521e+115
aa = 6.0521e+115
aa = -8.5590e+115
aa = 8.5590e+115
aa = -1.2104e+116
aa = 1.2104e+116
aa = -1.7118e+116
aa = 1.7118e+116
aa = -2.4209e+116
aa = 2.4209e+116
aa = -3.4236e+116
aa = 3.4236e+116
aa = -4.8417e+116
aa = 4.8417e+116
aa = -6.8472e+116
aa = 6.8472e+116
aa = -9.6834e+116
aa = 9.6834e+116
aa = -1.3694e+117
aa = 1.3694e+117
aa = -1.9367e+117
aa = 1.9367e+117
aa = -2.7389e+117
aa = 2.7389e+117
aa = -3.8734e+117
aa = 3.8734e+117
aa = -5.4778e+117
aa = 5.4778e+117
aa = -7.7467e+117
aa = 7.7467e+117
aa = -1.0956e+118
aa = 1.0956e+118
aa = -1.5493e+118
aa = 1.5493e+118
aa = -2.1911e+118
aa = 2.1911e+118
aa = -3.0987e+118
aa = 3.0987e+118
aa = -4.3822e+118
aa = 4.3822e+118
aa = -6.1974e+118
aa = 6.1974e+118
aa = -8.7644e+118
aa = 8.7644e+118
aa = -1.2395e+119
aa = 1.2395e+119
aa = -1.7529e+119
aa = 1.7529e+119
aa = -2.4790e+119
aa = 2.4790e+119
aa = -3.5058e+119
aa = 3.5058e+119
aa = -4.9579e+119
aa = 4.9579e+119
aa = -7.0116e+119
aa = 7.0116e+119
aa = -9.9158e+119
aa = 9.9158e+119
aa = -1.4023e+120
aa = 1.4023e+120
aa = -1.9832e+120
aa = 1.9832e+120
aa = -2.8046e+120
aa = 2.8046e+120
aa = -3.9663e+120
aa = 3.9663e+120
aa = -5.6092e+120
aa = 5.6092e+120
aa = -7.9327e+120
aa = 7.9327e+120
aa = -1.1218e+121
aa = 1.1218e+121
aa = -1.5865e+121
aa = 1.5865e+121
aa = -2.2437e+121
aa = 2.2437e+121
aa = -3.1731e+121
aa = 3.1731e+121
aa = -4.4874e+121
aa = 4.4874e+121
aa = -6.3461e+121
aa = 6.3461e+121
aa = -8.9748e+121
aa = 8.9748e+121
aa = -1.2692e+122
aa = 1.2692e+122
aa = -1.7950e+122
aa = 1.7950e+122
aa = -2.5385e+122
aa = 2.5385e+122
aa = -3.5899e+122
aa = 3.5899e+122
aa = -5.0769e+122
aa = 5.0769e+122
aa = -7.1798e+122
aa = 7.1798e+122
aa = -1.0154e+123
aa = 1.0154e+123
aa = -1.4360e+123
aa = 1.4360e+123
aa = -2.0308e+123
aa = 2.0308e+123
aa = -2.8719e+123
aa = 2.8719e+123
aa = -4.0615e+123
aa = 4.0615e+123
aa = -5.7439e+123
aa = 5.7439e+123
aa = -8.1231e+123
aa = 8.1231e+123
aa = -1.1488e+124
aa = 1.1488e+124
aa = -1.6246e+124
aa = 1.6246e+124
aa = -2.2975e+124
aa = 2.2975e+124
aa = -3.2492e+124
aa = 3.2492e+124
aa = -4.5951e+124
aa = 4.5951e+124
aa = -6.4984e+124
aa = 6.4984e+124
aa = -9.1902e+124
aa = 9.1902e+124
aa = -1.2997e+125
aa = 1.2997e+125
aa = -1.8380e+125
aa = 1.8380e+125
aa = -2.5994e+125
aa = 2.5994e+125
aa = -3.6761e+125
aa = 3.6761e+125
aa = -5.1988e+125
aa = 5.1988e+125
aa = -7.3522e+125
aa = 7.3522e+125
aa = -1.0398e+126
aa = 1.0398e+126
aa = -1.4704e+126
aa = 1.4704e+126
aa = -2.0795e+126
aa = 2.0795e+126
aa = -2.9409e+126
aa = 2.9409e+126
aa = -4.1590e+126
aa = 4.1590e+126
aa = -5.8817e+126
aa = 5.8817e+126
aa = -8.3180e+126
aa = 8.3180e+126
aa = -1.1763e+127
aa = 1.1763e+127
aa = -1.6636e+127
aa = 1.6636e+127
aa = -2.3527e+127
aa = 2.3527e+127
aa = -3.3272e+127
aa = 3.3272e+127
aa = -4.7054e+127
aa = 4.7054e+127
aa = -6.6544e+127
aa = 6.6544e+127
aa = -9.4108e+127
aa = 9.4108e+127
aa = -1.3309e+128
aa = 1.3309e+128
aa = -1.8822e+128
aa = 1.8822e+128
aa = -2.6618e+128
aa = 2.6618e+128
aa = -3.7643e+128
aa = 3.7643e+128
aa = -5.3235e+128
aa = 5.3235e+128
aa = -7.5286e+128
aa = 7.5286e+128
aa = -1.0647e+129
aa = 1.0647e+129
aa = -1.5057e+129
aa = 1.5057e+129
aa = -2.1294e+129
aa = 2.1294e+129
aa = -3.0114e+129
aa = 3.0114e+129
aa = -4.2588e+129
aa = 4.2588e+129
aa = -6.0229e+129
aa = 6.0229e+129
aa = -8.5176e+129
aa = 8.5176e+129
aa = -1.2046e+130
aa = 1.2046e+130
aa = -1.7035e+130
aa = 1.7035e+130
aa = -2.4092e+130
aa = 2.4092e+130
aa = -3.4071e+130
aa = 3.4071e+130
aa = -4.8183e+130
aa = 4.8183e+130
aa = -6.8141e+130
aa = 6.8141e+130
aa = -9.6366e+130
aa = 9.6366e+130
aa = -1.3628e+131
aa = 1.3628e+131
aa = -1.9273e+131
aa = 1.9273e+131
aa = -2.7256e+131
aa = 2.7256e+131
aa = -3.8546e+131
aa = 3.8546e+131
aa = -5.4513e+131
aa = 5.4513e+131
aa = -7.7093e+131
aa = 7.7093e+131
aa = -1.0903e+132
aa = 1.0903e+132
aa = -1.5419e+132
aa = 1.5419e+132
aa = -2.1805e+132
aa = 2.1805e+132
aa = -3.0837e+132
aa = 3.0837e+132
aa = -4.3610e+132
aa = 4.3610e+132
aa = -6.1674e+132
aa = 6.1674e+132
aa = -8.7221e+132
aa = 8.7221e+132
aa = -1.2335e+133
aa = 1.2335e+133
aa = -1.7444e+133
aa = 1.7444e+133
aa = -2.4670e+133
aa = 2.4670e+133
aa = -3.4888e+133
aa = 3.4888e+133
aa = -4.9339e+133
aa = 4.9339e+133
aa = -6.9777e+133
aa = 6.9777e+133
aa = -9.8679e+133
aa = 9.8679e+133
aa = -1.3955e+134
aa = 1.3955e+134
aa = -1.9736e+134
aa = 1.9736e+134
aa = -2.7911e+134
aa = 2.7911e+134
aa = -3.9472e+134
aa = 3.9472e+134
aa = -5.5821e+134
aa = 5.5821e+134
aa = -7.8943e+134
aa = 7.8943e+134
aa = -1.1164e+135
aa = 1.1164e+135
aa = -1.5789e+135
aa = 1.5789e+135
aa = -2.2328e+135
aa = 2.2328e+135
aa = -3.1577e+135
aa = 3.1577e+135
aa = -4.4657e+135
aa = 4.4657e+135
aa = -6.3154e+135
aa = 6.3154e+135
aa = -8.9314e+135
aa = 8.9314e+135
aa = -1.2631e+136
aa = 1.2631e+136
aa = -1.7863e+136
aa = 1.7863e+136
aa = -2.5262e+136
aa = 2.5262e+136
aa = -3.5726e+136
aa = 3.5726e+136
aa = -5.0524e+136
aa = 5.0524e+136
aa = -7.1451e+136
aa = 7.1451e+136
aa = -1.0105e+137
aa = 1.0105e+137
aa = -1.4290e+137
aa = 1.4290e+137
aa = -2.0209e+137
aa = 2.0209e+137
aa = -2.8580e+137
aa = 2.8580e+137
aa = -4.0419e+137
aa = 4.0419e+137
aa = -5.7161e+137
aa = 5.7161e+137
aa = -8.0838e+137
aa = 8.0838e+137
aa = -1.1432e+138
aa = 1.1432e+138
aa = -1.6168e+138
aa = 1.6168e+138
aa = -2.2864e+138
aa = 2.2864e+138
aa = -3.2335e+138
aa = 3.2335e+138
aa = -4.5729e+138
aa = 4.5729e+138
aa = -6.4670e+138
aa = 6.4670e+138
aa = -9.1457e+138
aa = 9.1457e+138
aa = -1.2934e+139
aa = 1.2934e+139
aa = -1.8291e+139
aa = 1.8291e+139
aa = -2.5868e+139
aa = 2.5868e+139
aa = -3.6583e+139
aa = 3.6583e+139
aa = -5.1736e+139
aa = 5.1736e+139
aa = -7.3166e+139
aa = 7.3166e+139
aa = -1.0347e+140
aa = 1.0347e+140
aa = -1.4633e+140
aa = 1.4633e+140
aa = -2.0694e+140
aa = 2.0694e+140
aa = -2.9266e+140
aa = 2.9266e+140
aa = -4.1389e+140
aa = 4.1389e+140
aa = -5.8533e+140
aa = 5.8533e+140
aa = -8.2778e+140
aa = 8.2778e+140
aa = -1.1707e+141
aa = 1.1707e+141
aa = -1.6556e+141
aa = 1.6556e+141
aa = -2.3413e+141
aa = 2.3413e+141
aa = -3.3111e+141
aa = 3.3111e+141
aa = -4.6826e+141
aa = 4.6826e+141
aa = -6.6222e+141
aa = 6.6222e+141
aa = -9.3652e+141
aa = 9.3652e+141
aa = -1.3244e+142
aa = 1.3244e+142
aa = -1.8730e+142
aa = 1.8730e+142
aa = -2.6489e+142
aa = 2.6489e+142
aa = -3.7461e+142
aa = 3.7461e+142
aa = -5.2978e+142
aa = 5.2978e+142
aa = -7.4922e+142
aa = 7.4922e+142
aa = -1.0596e+143
aa = 1.0596e+143
aa = -1.4984e+143
aa = 1.4984e+143
aa = -2.1191e+143
aa = 2.1191e+143
aa = -2.9969e+143
aa = 2.9969e+143
aa = -4.2382e+143
aa = 4.2382e+143
aa = -5.9938e+143
aa = 5.9938e+143
aa = -8.4765e+143
aa = 8.4765e+143
aa = -1.1988e+144
aa = 1.1988e+144
aa = -1.6953e+144
aa = 1.6953e+144
aa = -2.3975e+144
aa = 2.3975e+144
aa = -3.3906e+144
aa = 3.3906e+144
aa = -4.7950e+144
aa = 4.7950e+144
aa = -6.7812e+144
aa = 6.7812e+144
aa = -9.5900e+144
aa = 9.5900e+144
aa = -1.3562e+145
aa = 1.3562e+145
aa = -1.9180e+145
aa = 1.9180e+145
aa = -2.7125e+145
aa = 2.7125e+145
aa = -3.8360e+145
aa = 3.8360e+145
aa = -5.4249e+145
aa = 5.4249e+145
aa = -7.6720e+145
aa = 7.6720e+145
aa = -1.0850e+146
aa = 1.0850e+146
aa = -1.5344e+146
aa = 1.5344e+146
aa = -2.1700e+146
aa = 2.1700e+146
aa = -3.0688e+146
aa = 3.0688e+146
aa = -4.3399e+146
aa = 4.3399e+146
aa = -6.1376e+146
aa = 6.1376e+146
aa = -8.6799e+146
aa = 8.6799e+146
aa = -1.2275e+147
aa = 1.2275e+147
aa = -1.7360e+147
aa = 1.7360e+147
aa = -2.4550e+147
aa = 2.4550e+147
aa = -3.4720e+147
aa = 3.4720e+147
aa = -4.9101e+147
aa = 4.9101e+147
aa = -6.9439e+147
aa = 6.9439e+147
aa = -9.8202e+147
aa = 9.8202e+147
aa = -1.3888e+148
aa = 1.3888e+148
aa = -1.9640e+148
aa = 1.9640e+148
aa = -2.7776e+148
aa = 2.7776e+148
aa = -3.9281e+148
aa = 3.9281e+148
aa = -5.5551e+148
aa = 5.5551e+148
aa = -7.8561e+148
aa = 7.8561e+148
aa = -1.1110e+149
aa = 1.1110e+149
aa = -1.5712e+149
aa = 1.5712e+149
aa = -2.2221e+149
aa = 2.2221e+149
aa = -3.1425e+149
aa = 3.1425e+149
aa = -4.4441e+149
aa = 4.4441e+149
aa = -6.2849e+149
aa = 6.2849e+149
aa = -8.8882e+149
aa = 8.8882e+149
aa = -1.2570e+150
aa = 1.2570e+150
aa = -1.7776e+150
aa = 1.7776e+150
aa = -2.5140e+150
aa = 2.5140e+150
aa = -3.5553e+150
aa = 3.5553e+150
aa = -5.0279e+150
aa = 5.0279e+150
aa = -7.1106e+150
aa = 7.1106e+150
aa = -1.0056e+151
aa = 1.0056e+151
aa = -1.4221e+151
aa = 1.4221e+151
aa = -2.0112e+151
aa = 2.0112e+151
aa = -2.8442e+151
aa = 2.8442e+151
aa = -4.0223e+151
aa = 4.0223e+151
aa = -5.6885e+151
aa = 5.6885e+151
aa = -8.0447e+151
aa = 8.0447e+151
aa = -1.1377e+152
aa = 1.1377e+152
aa = -1.6089e+152
aa = 1.6089e+152
aa = -2.2754e+152
aa = 2.2754e+152
aa = -3.2179e+152
aa = 3.2179e+152
aa = -4.5508e+152
aa = 4.5508e+152
aa = -6.4357e+152
aa = 6.4357e+152
aa = -9.1015e+152
aa = 9.1015e+152
aa = -1.2871e+153
aa = 1.2871e+153
aa = -1.8203e+153
aa = 1.8203e+153
aa = -2.5743e+153
aa = 2.5743e+153
aa = -3.6406e+153
aa = 3.6406e+153
aa = -5.1486e+153
aa = 5.1486e+153
aa = -7.2812e+153
aa = 7.2812e+153
aa = -1.0297e+154
aa = 1.0297e+154
aa = -1.4562e+154
aa = 1.4562e+154
aa = -2.0594e+154
aa = 2.0594e+154
aa = -2.9125e+154
aa = 2.9125e+154
aa = -4.1189e+154
aa = 4.1189e+154
aa = -5.8250e+154
aa = 5.8250e+154
aa = -8.2378e+154
aa = 8.2378e+154
aa = -1.1650e+155
aa = 1.1650e+155
aa = -1.6476e+155
aa = 1.6476e+155
aa = -2.3300e+155
aa = 2.3300e+155
aa = -3.2951e+155
aa = 3.2951e+155
aa = -4.6600e+155
aa = 4.6600e+155
aa = -6.5902e+155
aa = 6.5902e+155
aa = -9.3200e+155
aa = 9.3200e+155
aa = -1.3180e+156
aa = 1.3180e+156
aa = -1.8640e+156
aa = 1.8640e+156
aa = -2.6361e+156
aa = 2.6361e+156
aa = -3.7280e+156
aa = 3.7280e+156
aa = -5.2722e+156
aa = 5.2722e+156
aa = -7.4560e+156
aa = 7.4560e+156
aa = -1.0544e+157
aa = 1.0544e+157
aa = -1.4912e+157
aa = 1.4912e+157
aa = -2.1089e+157
aa = 2.1089e+157
aa = -2.9824e+157
aa = 2.9824e+157
aa = -4.2177e+157
aa = 4.2177e+157
aa = -5.9648e+157
aa = 5.9648e+157
aa = -8.4355e+157
aa = 8.4355e+157
aa = -1.1930e+158
aa = 1.1930e+158
aa = -1.6871e+158
aa = 1.6871e+158
aa = -2.3859e+158
aa = 2.3859e+158
aa = -3.3742e+158
aa = 3.3742e+158
aa = -4.7718e+158
aa = 4.7718e+158
aa = -6.7484e+158
aa = 6.7484e+158
aa = -9.5436e+158
aa = 9.5436e+158
aa = -1.3497e+159
aa = 1.3497e+159
aa = -1.9087e+159
aa = 1.9087e+159
aa = -2.6993e+159
aa = 2.6993e+159
aa = -3.8175e+159
aa = 3.8175e+159
aa = -5.3987e+159
aa = 5.3987e+159
aa = -7.6349e+159
aa = 7.6349e+159
aa = -1.0797e+160
aa = 1.0797e+160
aa = -1.5270e+160
aa = 1.5270e+160
aa = -2.1595e+160
aa = 2.1595e+160
aa = -3.0540e+160
aa = 3.0540e+160
aa = -4.3190e+160
aa = 4.3190e+160
aa = -6.1079e+160
aa = 6.1079e+160
aa = -8.6379e+160
aa = 8.6379e+160
aa = -1.2216e+161
aa = 1.2216e+161
aa = -1.7276e+161
aa = 1.7276e+161
aa = -2.4432e+161
aa = 2.4432e+161
aa = -3.4552e+161
aa = 3.4552e+161
aa = -4.8863e+161
aa = 4.8863e+161
aa = -6.9103e+161
aa = 6.9103e+161
aa = -9.7727e+161
aa = 9.7727e+161
aa = -1.3821e+162
aa = 1.3821e+162
aa = -1.9545e+162
aa = 1.9545e+162
aa = -2.7641e+162
aa = 2.7641e+162
aa = -3.9091e+162
aa = 3.9091e+162
aa = -5.5283e+162
aa = 5.5283e+162
aa = -7.8181e+162
aa = 7.8181e+162
aa = -1.1057e+163
aa = 1.1057e+163
aa = -1.5636e+163
aa = 1.5636e+163
aa = -2.2113e+163
aa = 2.2113e+163
aa = -3.1273e+163
aa = 3.1273e+163
aa = -4.4226e+163
aa = 4.4226e+163
aa = -6.2545e+163
aa = 6.2545e+163
aa = -8.8452e+163
aa = 8.8452e+163
aa = -1.2509e+164
aa = 1.2509e+164
aa = -1.7690e+164
aa = 1.7690e+164
aa = -2.5018e+164
aa = 2.5018e+164
aa = -3.5381e+164
aa = 3.5381e+164
aa = -5.0036e+164
aa = 5.0036e+164
aa = -7.0762e+164
aa = 7.0762e+164
aa = -1.0007e+165
aa = 1.0007e+165
aa = -1.4152e+165
aa = 1.4152e+165
aa = -2.0014e+165
aa = 2.0014e+165
aa = -2.8305e+165
aa = 2.8305e+165
aa = -4.0029e+165
aa = 4.0029e+165
aa = -5.6609e+165
aa = 5.6609e+165
aa = -8.0058e+165
aa = 8.0058e+165
aa = -1.1322e+166
aa = 1.1322e+166
aa = -1.6012e+166
aa = 1.6012e+166
aa = -2.2644e+166
aa = 2.2644e+166
aa = -3.2023e+166
aa = 3.2023e+166
aa = -4.5288e+166
aa = 4.5288e+166
aa = -6.4046e+166
aa = 6.4046e+166
aa = -9.0575e+166
aa = 9.0575e+166
aa = -1.2809e+167
aa = 1.2809e+167
aa = -1.8115e+167
aa = 1.8115e+167
aa = -2.5619e+167
aa = 2.5619e+167
aa = -3.6230e+167
aa = 3.6230e+167
aa = -5.1237e+167
aa = 5.1237e+167
aa = -7.2460e+167
aa = 7.2460e+167
aa = -1.0247e+168
aa = 1.0247e+168
aa = -1.4492e+168
aa = 1.4492e+168
aa = -2.0495e+168
aa = 2.0495e+168
aa = -2.8984e+168
aa = 2.8984e+168
aa = -4.0990e+168
aa = 4.0990e+168
aa = -5.7968e+168
aa = 5.7968e+168
aa = -8.1979e+168
aa = 8.1979e+168
aa = -1.1594e+169
aa = 1.1594e+169
aa = -1.6396e+169
aa = 1.6396e+169
aa = -2.3187e+169
aa = 2.3187e+169
aa = -3.2792e+169
aa = 3.2792e+169
aa = -4.6374e+169
aa = 4.6374e+169
aa = -6.5583e+169
aa = 6.5583e+169
aa = -9.2749e+169
aa = 9.2749e+169
aa = -1.3117e+170
aa = 1.3117e+170
aa = -1.8550e+170
aa = 1.8550e+170
aa = -2.6233e+170
aa = 2.6233e+170
aa = -3.7100e+170
aa = 3.7100e+170
aa = -5.2467e+170
aa = 5.2467e+170
aa = -7.4199e+170
aa = 7.4199e+170
aa = -1.0493e+171
aa = 1.0493e+171
aa = -1.4840e+171
aa = 1.4840e+171
aa = -2.0987e+171
aa = 2.0987e+171
aa = -2.9680e+171
aa = 2.9680e+171
aa = -4.1973e+171
aa = 4.1973e+171
aa = -5.9359e+171
aa = 5.9359e+171
aa = -8.3947e+171
aa = 8.3947e+171
aa = -1.1872e+172
aa = 1.1872e+172
aa = -1.6789e+172
aa = 1.6789e+172
aa = -2.3744e+172
aa = 2.3744e+172
aa = -3.3579e+172
aa = 3.3579e+172
aa = -4.7487e+172
aa = 4.7487e+172
aa = -6.7157e+172
aa = 6.7157e+172
aa = -9.4975e+172
aa = 9.4975e+172
aa = -1.3431e+173
aa = 1.3431e+173
aa = -1.8995e+173
aa = 1.8995e+173
aa = -2.6863e+173
aa = 2.6863e+173
aa = -3.7990e+173
aa = 3.7990e+173
aa = -5.3726e+173
aa = 5.3726e+173
aa = -7.5980e+173
aa = 7.5980e+173
aa = -1.0745e+174
aa = 1.0745e+174
aa = -1.5196e+174
aa = 1.5196e+174
aa = -2.1490e+174
aa = 2.1490e+174
aa = -3.0392e+174
aa = 3.0392e+174
aa = -4.2981e+174
aa = 4.2981e+174
aa = -6.0784e+174
aa = 6.0784e+174
aa = -8.5961e+174
aa = 8.5961e+174
aa = -1.2157e+175
aa = 1.2157e+175
aa = -1.7192e+175
aa = 1.7192e+175
aa = -2.4314e+175
aa = 2.4314e+175
aa = -3.4385e+175
aa = 3.4385e+175
aa = -4.8627e+175
aa = 4.8627e+175
aa = -6.8769e+175
aa = 6.8769e+175
aa = -9.7254e+175
aa = 9.7254e+175
aa = -1.3754e+176
aa = 1.3754e+176
aa = -1.9451e+176
aa = 1.9451e+176
aa = -2.7508e+176
aa = 2.7508e+176
aa = -3.8902e+176
aa = 3.8902e+176
aa = -5.5015e+176
aa = 5.5015e+176
aa = -7.7803e+176
aa = 7.7803e+176
aa = -1.1003e+177
aa = 1.1003e+177
aa = -1.5561e+177
aa = 1.5561e+177
aa = -2.2006e+177
aa = 2.2006e+177
aa = -3.1121e+177
aa = 3.1121e+177
aa = -4.4012e+177
aa = 4.4012e+177
aa = -6.2243e+177
aa = 6.2243e+177
aa = -8.8025e+177
aa = 8.8025e+177
aa = -1.2449e+178
aa = 1.2449e+178
aa = -1.7605e+178
aa = 1.7605e+178
aa = -2.4897e+178
aa = 2.4897e+178
aa = -3.5210e+178
aa = 3.5210e+178
aa = -4.9794e+178
aa = 4.9794e+178
aa = -7.0420e+178
aa = 7.0420e+178
aa = -9.9588e+178
aa = 9.9588e+178
aa = -1.4084e+179
aa = 1.4084e+179
aa = -1.9918e+179
aa = 1.9918e+179
aa = -2.8168e+179
aa = 2.8168e+179
aa = -3.9835e+179
aa = 3.9835e+179
aa = -5.6336e+179
aa = 5.6336e+179
aa = -7.9671e+179
aa = 7.9671e+179
aa = -1.1267e+180
aa = 1.1267e+180
aa = -1.5934e+180
aa = 1.5934e+180
aa = -2.2534e+180
aa = 2.2534e+180
aa = -3.1868e+180
aa = 3.1868e+180
aa = -4.5069e+180
aa = 4.5069e+180
aa = -6.3737e+180
aa = 6.3737e+180
aa = -9.0137e+180
aa = 9.0137e+180
aa = -1.2747e+181
aa = 1.2747e+181
aa = -1.8027e+181
aa = 1.8027e+181
aa = -2.5495e+181
aa = 2.5495e+181
aa = -3.6055e+181
aa = 3.6055e+181
aa = -5.0989e+181
aa = 5.0989e+181
aa = -7.2110e+181
aa = 7.2110e+181
aa = -1.0198e+182
aa = 1.0198e+182
aa = -1.4422e+182
aa = 1.4422e+182
aa = -2.0396e+182
aa = 2.0396e+182
aa = -2.8844e+182
aa = 2.8844e+182
aa = -4.0791e+182
aa = 4.0791e+182
aa = -5.7688e+182
aa = 5.7688e+182
aa = -8.1583e+182
aa = 8.1583e+182
aa = -1.1538e+183
aa = 1.1538e+183
aa = -1.6317e+183
aa = 1.6317e+183
aa = -2.3075e+183
aa = 2.3075e+183
aa = -3.2633e+183
aa = 3.2633e+183
aa = -4.6150e+183
aa = 4.6150e+183
aa = -6.5266e+183
aa = 6.5266e+183
aa = -9.2300e+183
aa = 9.2300e+183
aa = -1.3053e+184
aa = 1.3053e+184
aa = -1.8460e+184
aa = 1.8460e+184
aa = -2.6106e+184
aa = 2.6106e+184
aa = -3.6920e+184
aa = 3.6920e+184
aa = -5.2213e+184
aa = 5.2213e+184
aa = -7.3840e+184
aa = 7.3840e+184
aa = -1.0443e+185
aa = 1.0443e+185
aa = -1.4768e+185
aa = 1.4768e+185
aa = -2.0885e+185
aa = 2.0885e+185
aa = -2.9536e+185
aa = 2.9536e+185
aa = -4.1770e+185
aa = 4.1770e+185
aa = -5.9072e+185
aa = 5.9072e+185
aa = -8.3541e+185
aa = 8.3541e+185
aa = -1.1814e+186
aa = 1.1814e+186
aa = -1.6708e+186
aa = 1.6708e+186
aa = -2.3629e+186
aa = 2.3629e+186
aa = -3.3416e+186
aa = 3.3416e+186
aa = -4.7258e+186
aa = 4.7258e+186
aa = -6.6833e+186
aa = 6.6833e+186
aa = -9.4516e+186
aa = 9.4516e+186
aa = -1.3367e+187
aa = 1.3367e+187
aa = -1.8903e+187
aa = 1.8903e+187
aa = -2.6733e+187
aa = 2.6733e+187
aa = -3.7806e+187
aa = 3.7806e+187
aa = -5.3466e+187
aa = 5.3466e+187
aa = -7.5612e+187
aa = 7.5612e+187
aa = -1.0693e+188
aa = 1.0693e+188
aa = -1.5122e+188
aa = 1.5122e+188
aa = -2.1386e+188
aa = 2.1386e+188
aa = -3.0245e+188
aa = 3.0245e+188
aa = -4.2773e+188
aa = 4.2773e+188
aa = -6.0490e+188
aa = 6.0490e+188
aa = -8.5546e+188
aa = 8.5546e+188
aa = -1.2098e+189
aa = 1.2098e+189
aa = -1.7109e+189
aa = 1.7109e+189
aa = -2.4196e+189
aa = 2.4196e+189
aa = -3.4218e+189
aa = 3.4218e+189
aa = -4.8392e+189
aa = 4.8392e+189
aa = -6.8437e+189
aa = 6.8437e+189
aa = -9.6784e+189
aa = 9.6784e+189
aa = -1.3687e+190
aa = 1.3687e+190
aa = -1.9357e+190
aa = 1.9357e+190
aa = -2.7375e+190
aa = 2.7375e+190
aa = -3.8714e+190
aa = 3.8714e+190
aa = -5.4749e+190
aa = 5.4749e+190
aa = -7.7427e+190
aa = 7.7427e+190
aa = -1.0950e+191
aa = 1.0950e+191
aa = -1.5485e+191
aa = 1.5485e+191
aa = -2.1900e+191
aa = 2.1900e+191
aa = -3.0971e+191
aa = 3.0971e+191
aa = -4.3799e+191
aa = 4.3799e+191
aa = -6.1942e+191
aa = 6.1942e+191
aa = -8.7599e+191
aa = 8.7599e+191
aa = -1.2388e+192
aa = 1.2388e+192
aa = -1.7520e+192
aa = 1.7520e+192
aa = -2.4777e+192
aa = 2.4777e+192
aa = -3.5040e+192
aa = 3.5040e+192
aa = -4.9553e+192
aa = 4.9553e+192
aa = -7.0079e+192
aa = 7.0079e+192
aa = -9.9107e+192
aa = 9.9107e+192
aa = -1.4016e+193
aa = 1.4016e+193
aa = -1.9821e+193
aa = 1.9821e+193
aa = -2.8032e+193
aa = 2.8032e+193
aa = -3.9643e+193
aa = 3.9643e+193
aa = -5.6063e+193
aa = 5.6063e+193
aa = -7.9285e+193
aa = 7.9285e+193
aa = -1.1213e+194
aa = 1.1213e+194
aa = -1.5857e+194
aa = 1.5857e+194
aa = -2.2425e+194
aa = 2.2425e+194
aa = -3.1714e+194
aa = 3.1714e+194
aa = -4.4851e+194
aa = 4.4851e+194
aa = -6.3428e+194
aa = 6.3428e+194
aa = -8.9701e+194
aa = 8.9701e+194
aa = -1.2686e+195
aa = 1.2686e+195
aa = -1.7940e+195
aa = 1.7940e+195
aa = -2.5371e+195
aa = 2.5371e+195
aa = -3.5880e+195
aa = 3.5880e+195
aa = -5.0743e+195
aa = 5.0743e+195
aa = -7.1761e+195
aa = 7.1761e+195
aa = -1.0149e+196
aa = 1.0149e+196
aa = -1.4352e+196
aa = 1.4352e+196
aa = -2.0297e+196
aa = 2.0297e+196
aa = -2.8704e+196
aa = 2.8704e+196
aa = -4.0594e+196
aa = 4.0594e+196
aa = -5.7409e+196
aa = 5.7409e+196
aa = -8.1188e+196
aa = 8.1188e+196
aa = -1.1482e+197
aa = 1.1482e+197
aa = -1.6238e+197
aa = 1.6238e+197
aa = -2.2964e+197
aa = 2.2964e+197
aa = -3.2475e+197
aa = 3.2475e+197
aa = -4.5927e+197
aa = 4.5927e+197
aa = -6.4951e+197
aa = 6.4951e+197
aa = -9.1854e+197
aa = 9.1854e+197
aa = -1.2990e+198
aa = 1.2990e+198
aa = -1.8371e+198
aa = 1.8371e+198
aa = -2.5980e+198
aa = 2.5980e+198
aa = -3.6742e+198
aa = 3.6742e+198
aa = -5.1961e+198
aa = 5.1961e+198
aa = -7.3483e+198
aa = 7.3483e+198
aa = -1.0392e+199
aa = 1.0392e+199
aa = -1.4697e+199
aa = 1.4697e+199
aa = -2.0784e+199
aa = 2.0784e+199
aa = -2.9393e+199
aa = 2.9393e+199
aa = -4.1568e+199
aa = 4.1568e+199
aa = -5.8787e+199
aa = 5.8787e+199
aa = -8.3137e+199
aa = 8.3137e+199
aa = -1.1757e+200
aa = 1.1757e+200
aa = -1.6627e+200
aa = 1.6627e+200
aa = -2.3515e+200
aa = 2.3515e+200
aa = -3.3255e+200
aa = 3.3255e+200
aa = -4.7029e+200
aa = 4.7029e+200
aa = -6.6509e+200
aa = 6.6509e+200
aa = -9.4059e+200
aa = 9.4059e+200
aa = -1.3302e+201
aa = 1.3302e+201
aa = -1.8812e+201
aa = 1.8812e+201
aa = -2.6604e+201
aa = 2.6604e+201
aa = -3.7623e+201
aa = 3.7623e+201
aa = -5.3208e+201
aa = 5.3208e+201
aa = -7.5247e+201
aa = 7.5247e+201
aa = -1.0642e+202
aa = 1.0642e+202
aa = -1.5049e+202
aa = 1.5049e+202
aa = -2.1283e+202
aa = 2.1283e+202
aa = -3.0099e+202
aa = 3.0099e+202
aa = -4.2566e+202
aa = 4.2566e+202
aa = -6.0197e+202
aa = 6.0197e+202
aa = -8.5132e+202
aa = 8.5132e+202
aa = -1.2039e+203
aa = 1.2039e+203
aa = -1.7026e+203
aa = 1.7026e+203
aa = -2.4079e+203
aa = 2.4079e+203
aa = -3.4053e+203
aa = 3.4053e+203
aa = -4.8158e+203
aa = 4.8158e+203
aa = -6.8106e+203
aa = 6.8106e+203
aa = -9.6316e+203
aa = 9.6316e+203
aa = -1.3621e+204
aa = 1.3621e+204
aa = -1.9263e+204
aa = 1.9263e+204
aa = -2.7242e+204
aa = 2.7242e+204
aa = -3.8526e+204
aa = 3.8526e+204
aa = -5.4485e+204
aa = 5.4485e+204
aa = -7.7053e+204
aa = 7.7053e+204
aa = -1.0897e+205
aa = 1.0897e+205
aa = -1.5411e+205
aa = 1.5411e+205
aa = -2.1794e+205
aa = 2.1794e+205
aa = -3.0821e+205
aa = 3.0821e+205
aa = -4.3588e+205
aa = 4.3588e+205
aa = -6.1642e+205
aa = 6.1642e+205
aa = -8.7175e+205
aa = 8.7175e+205
aa = -1.2328e+206
aa = 1.2328e+206
aa = -1.7435e+206
aa = 1.7435e+206
aa = -2.4657e+206
aa = 2.4657e+206
aa = -3.4870e+206
aa = 3.4870e+206
aa = -4.9314e+206
aa = 4.9314e+206
aa = -6.9740e+206
aa = 6.9740e+206
aa = -9.8628e+206
aa = 9.8628e+206
aa = -1.3948e+207
aa = 1.3948e+207
aa = -1.9726e+207
aa = 1.9726e+207
aa = -2.7896e+207
aa = 2.7896e+207
aa = -3.9451e+207
aa = 3.9451e+207
aa = -5.5792e+207
aa = 5.5792e+207
aa = -7.8902e+207
aa = 7.8902e+207
aa = -1.1158e+208
aa = 1.1158e+208
aa = -1.5780e+208
aa = 1.5780e+208
aa = -2.2317e+208
aa = 2.2317e+208
aa = -3.1561e+208
aa = 3.1561e+208
aa = -4.4634e+208
aa = 4.4634e+208
aa = -6.3122e+208
aa = 6.3122e+208
aa = -8.9267e+208
aa = 8.9267e+208
aa = -1.2624e+209
aa = 1.2624e+209
aa = -1.7853e+209
aa = 1.7853e+209
aa = -2.5249e+209
aa = 2.5249e+209
aa = -3.5707e+209
aa = 3.5707e+209
aa = -5.0497e+209
aa = 5.0497e+209
aa = -7.1414e+209
aa = 7.1414e+209
aa = -1.0099e+210
aa = 1.0099e+210
aa = -1.4283e+210
aa = 1.4283e+210
aa = -2.0199e+210
aa = 2.0199e+210
aa = -2.8566e+210
aa = 2.8566e+210
aa = -4.0398e+210
aa = 4.0398e+210
aa = -5.7131e+210
aa = 5.7131e+210
aa = -8.0796e+210
aa = 8.0796e+210
aa = -1.1426e+211
aa = 1.1426e+211
aa = -1.6159e+211
aa = 1.6159e+211
aa = -2.2852e+211
aa = 2.2852e+211
aa = -3.2318e+211
aa = 3.2318e+211
aa = -4.5705e+211
aa = 4.5705e+211
aa = -6.4637e+211
aa = 6.4637e+211
aa = -9.1410e+211
aa = 9.1410e+211
aa = -1.2927e+212
aa = 1.2927e+212
aa = -1.8282e+212
aa = 1.8282e+212
aa = -2.5855e+212
aa = 2.5855e+212
aa = -3.6564e+212
aa = 3.6564e+212
aa = -5.1709e+212
aa = 5.1709e+212
aa = -7.3128e+212
aa = 7.3128e+212
aa = -1.0342e+213
aa = 1.0342e+213
aa = -1.4626e+213
aa = 1.4626e+213
aa = -2.0684e+213
aa = 2.0684e+213
aa = -2.9251e+213
aa = 2.9251e+213
aa = -4.1367e+213
aa = 4.1367e+213
aa = -5.8502e+213
aa = 5.8502e+213
aa = -8.2735e+213
aa = 8.2735e+213
aa = -1.1700e+214
aa = 1.1700e+214
aa = -1.6547e+214
aa = 1.6547e+214
aa = -2.3401e+214
aa = 2.3401e+214
aa = -3.3094e+214
aa = 3.3094e+214
aa = -4.6802e+214
aa = 4.6802e+214
aa = -6.6188e+214
aa = 6.6188e+214
aa = -9.3604e+214
aa = 9.3604e+214
aa = -1.3238e+215
aa = 1.3238e+215
aa = -1.8721e+215
aa = 1.8721e+215
aa = -2.6475e+215
aa = 2.6475e+215
aa = -3.7441e+215
aa = 3.7441e+215
aa = -5.2950e+215
aa = 5.2950e+215
aa = -7.4883e+215
aa = 7.4883e+215
aa = -1.0590e+216
aa = 1.0590e+216
aa = -1.4977e+216
aa = 1.4977e+216
aa = -2.1180e+216
aa = 2.1180e+216
aa = -2.9953e+216
aa = 2.9953e+216
aa = -4.2360e+216
aa = 4.2360e+216
aa = -5.9906e+216
aa = 5.9906e+216
aa = -8.4720e+216
aa = 8.4720e+216
aa = -1.1981e+217
aa = 1.1981e+217
aa = -1.6944e+217
aa = 1.6944e+217
aa = -2.3963e+217
aa = 2.3963e+217
aa = -3.3888e+217
aa = 3.3888e+217
aa = -4.7925e+217
aa = 4.7925e+217
aa = -6.7776e+217
aa = 6.7776e+217
aa = -9.5850e+217
aa = 9.5850e+217
aa = -1.3555e+218
aa = 1.3555e+218
aa = -1.9170e+218
aa = 1.9170e+218
aa = -2.7111e+218
aa = 2.7111e+218
aa = -3.8340e+218
aa = 3.8340e+218
aa = -5.4221e+218
aa = 5.4221e+218
aa = -7.6680e+218
aa = 7.6680e+218
aa = -1.0844e+219
aa = 1.0844e+219
aa = -1.5336e+219
aa = 1.5336e+219
aa = -2.1688e+219
aa = 2.1688e+219
aa = -3.0672e+219
aa = 3.0672e+219
aa = -4.3377e+219
aa = 4.3377e+219
aa = -6.1344e+219
aa = 6.1344e+219
aa = -8.6754e+219
aa = 8.6754e+219
aa = -1.2269e+220
aa = 1.2269e+220
aa = -1.7351e+220
aa = 1.7351e+220
aa = -2.4538e+220
aa = 2.4538e+220
aa = -3.4701e+220
aa = 3.4701e+220
aa = -4.9075e+220
aa = 4.9075e+220
aa = -6.9403e+220
aa = 6.9403e+220
aa = -9.8151e+220
aa = 9.8151e+220
aa = -1.3881e+221
aa = 1.3881e+221
aa = -1.9630e+221
aa = 1.9630e+221
aa = -2.7761e+221
aa = 2.7761e+221
aa = -3.9260e+221
aa = 3.9260e+221
aa = -5.5522e+221
aa = 5.5522e+221
aa = -7.8520e+221
aa = 7.8520e+221
aa = -1.1104e+222
aa = 1.1104e+222
aa = -1.5704e+222
aa = 1.5704e+222
aa = -2.2209e+222
aa = 2.2209e+222
aa = -3.1408e+222
aa = 3.1408e+222
aa = -4.4418e+222
aa = 4.4418e+222
aa = -6.2816e+222
aa = 6.2816e+222
aa = -8.8836e+222
aa = 8.8836e+222
aa = -1.2563e+223
aa = 1.2563e+223
aa = -1.7767e+223
aa = 1.7767e+223
aa = -2.5127e+223
aa = 2.5127e+223
aa = -3.5534e+223
aa = 3.5534e+223
aa = -5.0253e+223
aa = 5.0253e+223
aa = -7.1069e+223
aa = 7.1069e+223
aa = -1.0051e+224
aa = 1.0051e+224
aa = -1.4214e+224
aa = 1.4214e+224
aa = -2.0101e+224
aa = 2.0101e+224
aa = -2.8427e+224
aa = 2.8427e+224
aa = -4.0202e+224
aa = 4.0202e+224
aa = -5.6855e+224
aa = 5.6855e+224
aa = -8.0405e+224
aa = 8.0405e+224
aa = -1.1371e+225
aa = 1.1371e+225
aa = -1.6081e+225
aa = 1.6081e+225
aa = -2.2742e+225
aa = 2.2742e+225
aa = -3.2162e+225
aa = 3.2162e+225
aa = -4.5484e+225
aa = 4.5484e+225
aa = -6.4324e+225
aa = 6.4324e+225
aa = -9.0968e+225
aa = 9.0968e+225
aa = -1.2865e+226
aa = 1.2865e+226
aa = -1.8194e+226
aa = 1.8194e+226
aa = -2.5730e+226
aa = 2.5730e+226
aa = -3.6387e+226
aa = 3.6387e+226
aa = -5.1459e+226
aa = 5.1459e+226
aa = -7.2774e+226
aa = 7.2774e+226
aa = -1.0292e+227
aa = 1.0292e+227
aa = -1.4555e+227
aa = 1.4555e+227
aa = -2.0584e+227
aa = 2.0584e+227
aa = -2.9110e+227
aa = 2.9110e+227
aa = -4.1167e+227
aa = 4.1167e+227
aa = -5.8219e+227
aa = 5.8219e+227
aa = -8.2335e+227
aa = 8.2335e+227
aa = -1.1644e+228
aa = 1.1644e+228
aa = -1.6467e+228
aa = 1.6467e+228
aa = -2.3288e+228
aa = 2.3288e+228
aa = -3.2934e+228
aa = 3.2934e+228
aa = -4.6576e+228
aa = 4.6576e+228
aa = -6.5868e+228
aa = 6.5868e+228
aa = -9.3151e+228
aa = 9.3151e+228
aa = -1.3174e+229
aa = 1.3174e+229
aa = -1.8630e+229
aa = 1.8630e+229
aa = -2.6347e+229
aa = 2.6347e+229
aa = -3.7260e+229
aa = 3.7260e+229
aa = -5.2694e+229
aa = 5.2694e+229
aa = -7.4521e+229
aa = 7.4521e+229
aa = -1.0539e+230
aa = 1.0539e+230
aa = -1.4904e+230
aa = 1.4904e+230
aa = -2.1078e+230
aa = 2.1078e+230
aa = -2.9808e+230
aa = 2.9808e+230
aa = -4.2155e+230
aa = 4.2155e+230
aa = -5.9617e+230
aa = 5.9617e+230
aa = -8.4311e+230
aa = 8.4311e+230
aa = -1.1923e+231
aa = 1.1923e+231
aa = -1.6862e+231
aa = 1.6862e+231
aa = -2.3847e+231
aa = 2.3847e+231
aa = -3.3724e+231
aa = 3.3724e+231
aa = -4.7693e+231
aa = 4.7693e+231
aa = -6.7449e+231
aa = 6.7449e+231
aa = -9.5387e+231
aa = 9.5387e+231
aa = -1.3490e+232
aa = 1.3490e+232
aa = -1.9077e+232
aa = 1.9077e+232
aa = -2.6979e+232
aa = 2.6979e+232
aa = -3.8155e+232
aa = 3.8155e+232
aa = -5.3959e+232
aa = 5.3959e+232
aa = -7.6309e+232
aa = 7.6309e+232
aa = -1.0792e+233
aa = 1.0792e+233
aa = -1.5262e+233
aa = 1.5262e+233
aa = -2.1584e+233
aa = 2.1584e+233
aa = -3.0524e+233
aa = 3.0524e+233
aa = -4.3167e+233
aa = 4.3167e+233
aa = -6.1047e+233
aa = 6.1047e+233
aa = -8.6334e+233
aa = 8.6334e+233
aa = -1.2209e+234
aa = 1.2209e+234
aa = -1.7267e+234
aa = 1.7267e+234
aa = -2.4419e+234
aa = 2.4419e+234
aa = -3.4534e+234
aa = 3.4534e+234
aa = -4.8838e+234
aa = 4.8838e+234
aa = -6.9067e+234
aa = 6.9067e+234
aa = -9.7676e+234
aa = 9.7676e+234
aa = -1.3813e+235
aa = 1.3813e+235
aa = -1.9535e+235
aa = 1.9535e+235
aa = -2.7627e+235
aa = 2.7627e+235
aa = -3.9070e+235
aa = 3.9070e+235
aa = -5.5254e+235
aa = 5.5254e+235
aa = -7.8141e+235
aa = 7.8141e+235
aa = -1.1051e+236
aa = 1.1051e+236
aa = -1.5628e+236
aa = 1.5628e+236
aa = -2.2102e+236
aa = 2.2102e+236
aa = -3.1256e+236
aa = 3.1256e+236
aa = -4.4203e+236
aa = 4.4203e+236
aa = -6.2513e+236
aa = 6.2513e+236
aa = -8.8406e+236
aa = 8.8406e+236
aa = -1.2503e+237
aa = 1.2503e+237
aa = -1.7681e+237
aa = 1.7681e+237
aa = -2.5005e+237
aa = 2.5005e+237
aa = -3.5362e+237
aa = 3.5362e+237
aa = -5.0010e+237
aa = 5.0010e+237
aa = -7.0725e+237
aa = 7.0725e+237
aa = -1.0002e+238
aa = 1.0002e+238
aa = -1.4145e+238
aa = 1.4145e+238
aa = -2.0004e+238
aa = 2.0004e+238
aa = -2.8290e+238
aa = 2.8290e+238
aa = -4.0008e+238
aa = 4.0008e+238
aa = -5.6580e+238
aa = 5.6580e+238
aa = -8.0016e+238
aa = 8.0016e+238
aa = -1.1316e+239
aa = 1.1316e+239
aa = -1.6003e+239
aa = 1.6003e+239
aa = -2.2632e+239
aa = 2.2632e+239
aa = -3.2006e+239
aa = 3.2006e+239
aa = -4.5264e+239
aa = 4.5264e+239
aa = -6.4013e+239
aa = 6.4013e+239
aa = -9.0528e+239
aa = 9.0528e+239
aa = -1.2803e+240
aa = 1.2803e+240
aa = -1.8106e+240
aa = 1.8106e+240
aa = -2.5605e+240
aa = 2.5605e+240
aa = -3.6211e+240
aa = 3.6211e+240
aa = -5.1210e+240
aa = 5.1210e+240
aa = -7.2422e+240
aa = 7.2422e+240
aa = -1.0242e+241
aa = 1.0242e+241
aa = -1.4484e+241
aa = 1.4484e+241
aa = -2.0484e+241
aa = 2.0484e+241
aa = -2.8969e+241
aa = 2.8969e+241
aa = -4.0968e+241
aa = 4.0968e+241
aa = -5.7938e+241
aa = 5.7938e+241
aa = -8.1937e+241
aa = 8.1937e+241
aa = -1.1588e+242
aa = 1.1588e+242
aa = -1.6387e+242
aa = 1.6387e+242
aa = -2.3175e+242
aa = 2.3175e+242
aa = -3.2775e+242
aa = 3.2775e+242
aa = -4.6350e+242
aa = 4.6350e+242
aa = -6.5549e+242
aa = 6.5549e+242
aa = -9.2701e+242
aa = 9.2701e+242
aa = -1.3110e+243
aa = 1.3110e+243
aa = -1.8540e+243
aa = 1.8540e+243
aa = -2.6220e+243
aa = 2.6220e+243
aa = -3.7080e+243
aa = 3.7080e+243
aa = -5.2439e+243
aa = 5.2439e+243
aa = -7.4161e+243
aa = 7.4161e+243
aa = -1.0488e+244
aa = 1.0488e+244
aa = -1.4832e+244
aa = 1.4832e+244
aa = -2.0976e+244
aa = 2.0976e+244
aa = -2.9664e+244
aa = 2.9664e+244
aa = -4.1952e+244
aa = 4.1952e+244
aa = -5.9328e+244
aa = 5.9328e+244
aa = -8.3903e+244
aa = 8.3903e+244
aa = -1.1866e+245
aa = 1.1866e+245
aa = -1.6781e+245
aa = 1.6781e+245
aa = -2.3731e+245
aa = 2.3731e+245
aa = -3.3561e+245
aa = 3.3561e+245
aa = -4.7463e+245
aa = 4.7463e+245
aa = -6.7122e+245
aa = 6.7122e+245
aa = -9.4925e+245
aa = 9.4925e+245
aa = -1.3424e+246
aa = 1.3424e+246
aa = -1.8985e+246
aa = 1.8985e+246
aa = -2.6849e+246
aa = 2.6849e+246
aa = -3.7970e+246
aa = 3.7970e+246
aa = -5.3698e+246
aa = 5.3698e+246
aa = -7.5940e+246
aa = 7.5940e+246
aa = -1.0740e+247
aa = 1.0740e+247
aa = -1.5188e+247
aa = 1.5188e+247
aa = -2.1479e+247
aa = 2.1479e+247
aa = -3.0376e+247
aa = 3.0376e+247
aa = -4.2958e+247
aa = 4.2958e+247
aa = -6.0752e+247
aa = 6.0752e+247
aa = -8.5917e+247
aa = 8.5917e+247
aa = -1.2150e+248
aa = 1.2150e+248
aa = -1.7183e+248
aa = 1.7183e+248
aa = -2.4301e+248
aa = 2.4301e+248
aa = -3.4367e+248
aa = 3.4367e+248
aa = -4.8602e+248
aa = 4.8602e+248
aa = -6.8733e+248
aa = 6.8733e+248
aa = -9.7204e+248
aa = 9.7204e+248
aa = -1.3747e+249
aa = 1.3747e+249
aa = -1.9441e+249
aa = 1.9441e+249
aa = -2.7493e+249
aa = 2.7493e+249
aa = -3.8881e+249
aa = 3.8881e+249
aa = -5.4987e+249
aa = 5.4987e+249
aa = -7.7763e+249
aa = 7.7763e+249
aa = -1.0997e+250
aa = 1.0997e+250
aa = -1.5553e+250
aa = 1.5553e+250
aa = -2.1995e+250
aa = 2.1995e+250
aa = -3.1105e+250
aa = 3.1105e+250
aa = -4.3989e+250
aa = 4.3989e+250
aa = -6.2210e+250
aa = 6.2210e+250
aa = -8.7979e+250
aa = 8.7979e+250
aa = -1.2442e+251
aa = 1.2442e+251
aa = -1.7596e+251
aa = 1.7596e+251
aa = -2.4884e+251
aa = 2.4884e+251
aa = -3.5191e+251
aa = 3.5191e+251
aa = -4.9768e+251
aa = 4.9768e+251
aa = -7.0383e+251
aa = 7.0383e+251
aa = -9.9537e+251
aa = 9.9537e+251
aa = -1.4077e+252
aa = 1.4077e+252
aa = -1.9907e+252
aa = 1.9907e+252
aa = -2.8153e+252
aa = 2.8153e+252
aa = -3.9815e+252
aa = 3.9815e+252
aa = -5.6306e+252
aa = 5.6306e+252
aa = -7.9629e+252
aa = 7.9629e+252
aa = -1.1261e+253
aa = 1.1261e+253
aa = -1.5926e+253
aa = 1.5926e+253
aa = -2.2523e+253
aa = 2.2523e+253
aa = -3.1852e+253
aa = 3.1852e+253
aa = -4.5045e+253
aa = 4.5045e+253
aa = -6.3703e+253
aa = 6.3703e+253
aa = -9.0090e+253
aa = 9.0090e+253
aa = -1.2741e+254
aa = 1.2741e+254
aa = -1.8018e+254
aa = 1.8018e+254
aa = -2.5481e+254
aa = 2.5481e+254
aa = -3.6036e+254
aa = 3.6036e+254
aa = -5.0963e+254
aa = 5.0963e+254
aa = -7.2072e+254
aa = 7.2072e+254
aa = -1.0193e+255
aa = 1.0193e+255
aa = -1.4414e+255
aa = 1.4414e+255
aa = -2.0385e+255
aa = 2.0385e+255
aa = -2.8829e+255
aa = 2.8829e+255
aa = -4.0770e+255
aa = 4.0770e+255
aa = -5.7658e+255
aa = 5.7658e+255
aa = -8.1540e+255
aa = 8.1540e+255
aa = -1.1532e+256
aa = 1.1532e+256
aa = -1.6308e+256
aa = 1.6308e+256
aa = -2.3063e+256
aa = 2.3063e+256
aa = -3.2616e+256
aa = 3.2616e+256
aa = -4.6126e+256
aa = 4.6126e+256
aa = -6.5232e+256
aa = 6.5232e+256
aa = -9.2252e+256
aa = 9.2252e+256
aa = -1.3046e+257
aa = 1.3046e+257
aa = -1.8450e+257
aa = 1.8450e+257
aa = -2.6093e+257
aa = 2.6093e+257
aa = -3.6901e+257
aa = 3.6901e+257
aa = -5.2186e+257
aa = 5.2186e+257
aa = -7.3802e+257
aa = 7.3802e+257
aa = -1.0437e+258
aa = 1.0437e+258
aa = -1.4760e+258
aa = 1.4760e+258
aa = -2.0874e+258
aa = 2.0874e+258
aa = -2.9521e+258
aa = 2.9521e+258
aa = -4.1749e+258
aa = 4.1749e+258
aa = -5.9042e+258
aa = 5.9042e+258
aa = -8.3497e+258
aa = 8.3497e+258
aa = -1.1808e+259
aa = 1.1808e+259
aa = -1.6699e+259
aa = 1.6699e+259
aa = -2.3617e+259
aa = 2.3617e+259
aa = -3.3399e+259
aa = 3.3399e+259
aa = -4.7233e+259
aa = 4.7233e+259
aa = -6.6798e+259
aa = 6.6798e+259
aa = -9.4466e+259
aa = 9.4466e+259
aa = -1.3360e+260
aa = 1.3360e+260
aa = -1.8893e+260
aa = 1.8893e+260
aa = -2.6719e+260
aa = 2.6719e+260
aa = -3.7787e+260
aa = 3.7787e+260
aa = -5.3438e+260
aa = 5.3438e+260
aa = -7.5573e+260
aa = 7.5573e+260
aa = -1.0688e+261
aa = 1.0688e+261
aa = -1.5115e+261
aa = 1.5115e+261
aa = -2.1375e+261
aa = 2.1375e+261
aa = -3.0229e+261
aa = 3.0229e+261
aa = -4.2751e+261
aa = 4.2751e+261
aa = -6.0459e+261
aa = 6.0459e+261
aa = -8.5501e+261
aa = 8.5501e+261
aa = -1.2092e+262
aa = 1.2092e+262
aa = -1.7100e+262
aa = 1.7100e+262
aa = -2.4183e+262
aa = 2.4183e+262
aa = -3.4200e+262
aa = 3.4200e+262
aa = -4.8367e+262
aa = 4.8367e+262
aa = -6.8401e+262
aa = 6.8401e+262
aa = -9.6734e+262
aa = 9.6734e+262
aa = -1.3680e+263
aa = 1.3680e+263
aa = -1.9347e+263
aa = 1.9347e+263
aa = -2.7360e+263
aa = 2.7360e+263
aa = -3.8693e+263
aa = 3.8693e+263
aa = -5.4721e+263
aa = 5.4721e+263
aa = -7.7387e+263
aa = 7.7387e+263
aa = -1.0944e+264
aa = 1.0944e+264
aa = -1.5477e+264
aa = 1.5477e+264
aa = -2.1888e+264
aa = 2.1888e+264
aa = -3.0955e+264
aa = 3.0955e+264
aa = -4.3777e+264
aa = 4.3777e+264
aa = -6.1910e+264
aa = 6.1910e+264
aa = -8.7553e+264
aa = 8.7553e+264
aa = -1.2382e+265
aa = 1.2382e+265
aa = -1.7511e+265
aa = 1.7511e+265
aa = -2.4764e+265
aa = 2.4764e+265
aa = -3.5021e+265
aa = 3.5021e+265
aa = -4.9528e+265
aa = 4.9528e+265
aa = -7.0043e+265
aa = 7.0043e+265
aa = -9.9055e+265
aa = 9.9055e+265
aa = -1.4009e+266
aa = 1.4009e+266
aa = -1.9811e+266
aa = 1.9811e+266
aa = -2.8017e+266
aa = 2.8017e+266
aa = -3.9622e+266
aa = 3.9622e+266
aa = -5.6034e+266
aa = 5.6034e+266
aa = -7.9244e+266
aa = 7.9244e+266
aa = -1.1207e+267
aa = 1.1207e+267
aa = -1.5849e+267
aa = 1.5849e+267
aa = -2.2414e+267
aa = 2.2414e+267
aa = -3.1698e+267
aa = 3.1698e+267
aa = -4.4827e+267
aa = 4.4827e+267
aa = -6.3395e+267
aa = 6.3395e+267
aa = -8.9655e+267
aa = 8.9655e+267
aa = -1.2679e+268
aa = 1.2679e+268
aa = -1.7931e+268
aa = 1.7931e+268
aa = -2.5358e+268
aa = 2.5358e+268
aa = -3.5862e+268
aa = 3.5862e+268
aa = -5.0716e+268
aa = 5.0716e+268
aa = -7.1724e+268
aa = 7.1724e+268
aa = -1.0143e+269
aa = 1.0143e+269
aa = -1.4345e+269
aa = 1.4345e+269
aa = -2.0287e+269
aa = 2.0287e+269
aa = -2.8689e+269
aa = 2.8689e+269
aa = -4.0573e+269
aa = 4.0573e+269
aa = -5.7379e+269
aa = 5.7379e+269
aa = -8.1146e+269
aa = 8.1146e+269
aa = -1.1476e+270
aa = 1.1476e+270
aa = -1.6229e+270
aa = 1.6229e+270
aa = -2.2952e+270
aa = 2.2952e+270
aa = -3.2458e+270
aa = 3.2458e+270
aa = -4.5903e+270
aa = 4.5903e+270
aa = -6.4917e+270
aa = 6.4917e+270
aa = -9.1806e+270
aa = 9.1806e+270
aa = -1.2983e+271
aa = 1.2983e+271
aa = -1.8361e+271
aa = 1.8361e+271
aa = -2.5967e+271
aa = 2.5967e+271
aa = -3.6723e+271
aa = 3.6723e+271
aa = -5.1933e+271
aa = 5.1933e+271
aa = -7.3445e+271
aa = 7.3445e+271
aa = -1.0387e+272
aa = 1.0387e+272
aa = -1.4689e+272
aa = 1.4689e+272
aa = -2.0773e+272
aa = 2.0773e+272
aa = -2.9378e+272
aa = 2.9378e+272
aa = -4.1547e+272
aa = 4.1547e+272
aa = -5.8756e+272
aa = 5.8756e+272
aa = -8.3094e+272
aa = 8.3094e+272
aa = -1.1751e+273
aa = 1.1751e+273
aa = -1.6619e+273
aa = 1.6619e+273
aa = -2.3502e+273
aa = 2.3502e+273
aa = -3.3237e+273
aa = 3.3237e+273
aa = -4.7005e+273
aa = 4.7005e+273
aa = -6.6475e+273
aa = 6.6475e+273
aa = -9.4010e+273
aa = 9.4010e+273
aa = -1.3295e+274
aa = 1.3295e+274
aa = -1.8802e+274
aa = 1.8802e+274
aa = -2.6590e+274
aa = 2.6590e+274
aa = -3.7604e+274
aa = 3.7604e+274
aa = -5.3180e+274
aa = 5.3180e+274
aa = -7.5208e+274
aa = 7.5208e+274
aa = -1.0636e+275
aa = 1.0636e+275
aa = -1.5042e+275
aa = 1.5042e+275
aa = -2.1272e+275
aa = 2.1272e+275
aa = -3.0083e+275
aa = 3.0083e+275
aa = -4.2544e+275
aa = 4.2544e+275
aa = -6.0166e+275
aa = 6.0166e+275
aa = -8.5088e+275
aa = 8.5088e+275
aa = -1.2033e+276
aa = 1.2033e+276
aa = -1.7018e+276
aa = 1.7018e+276
aa = -2.4066e+276
aa = 2.4066e+276
aa = -3.4035e+276
aa = 3.4035e+276
aa = -4.8133e+276
aa = 4.8133e+276
aa = -6.8070e+276
aa = 6.8070e+276
aa = -9.6266e+276
aa = 9.6266e+276
aa = -1.3614e+277
aa = 1.3614e+277
aa = -1.9253e+277
aa = 1.9253e+277
aa = -2.7228e+277
aa = 2.7228e+277
aa = -3.8506e+277
aa = 3.8506e+277
aa = -5.4456e+277
aa = 5.4456e+277
aa = -7.7013e+277
aa = 7.7013e+277
aa = -1.0891e+278
aa = 1.0891e+278
aa = -1.5403e+278
aa = 1.5403e+278
aa = -2.1782e+278
aa = 2.1782e+278
aa = -3.0805e+278
aa = 3.0805e+278
aa = -4.3565e+278
aa = 4.3565e+278
aa = -6.1610e+278
aa = 6.1610e+278
aa = -8.7130e+278
aa = 8.7130e+278
aa = -1.2322e+279
aa = 1.2322e+279
aa = -1.7426e+279
aa = 1.7426e+279
aa = -2.4644e+279
aa = 2.4644e+279
aa = -3.4852e+279
aa = 3.4852e+279
aa = -4.9288e+279
aa = 4.9288e+279
aa = -6.9704e+279
aa = 6.9704e+279
aa = -9.8576e+279
aa = 9.8576e+279
aa = -1.3941e+280
aa = 1.3941e+280
aa = -1.9715e+280
aa = 1.9715e+280
aa = -2.7882e+280
aa = 2.7882e+280
aa = -3.9430e+280
aa = 3.9430e+280
aa = -5.5763e+280
aa = 5.5763e+280
aa = -7.8861e+280
aa = 7.8861e+280
aa = -1.1153e+281
aa = 1.1153e+281
aa = -1.5772e+281
aa = 1.5772e+281
aa = -2.2305e+281
aa = 2.2305e+281
aa = -3.1544e+281
aa = 3.1544e+281
aa = -4.4611e+281
aa = 4.4611e+281
aa = -6.3089e+281
aa = 6.3089e+281
aa = -8.9221e+281
aa = 8.9221e+281
aa = -1.2618e+282
aa = 1.2618e+282
aa = -1.7844e+282
aa = 1.7844e+282
aa = -2.5236e+282
aa = 2.5236e+282
aa = -3.5688e+282
aa = 3.5688e+282
aa = -5.0471e+282
aa = 5.0471e+282
aa = -7.1377e+282
aa = 7.1377e+282
aa = -1.0094e+283
aa = 1.0094e+283
aa = -1.4275e+283
aa = 1.4275e+283
aa = -2.0188e+283
aa = 2.0188e+283
aa = -2.8551e+283
aa = 2.8551e+283
aa = -4.0377e+283
aa = 4.0377e+283
aa = -5.7101e+283
aa = 5.7101e+283
aa = -8.0754e+283
aa = 8.0754e+283
aa = -1.1420e+284
aa = 1.1420e+284
aa = -1.6151e+284
aa = 1.6151e+284
aa = -2.2841e+284
aa = 2.2841e+284
aa = -3.2301e+284
aa = 3.2301e+284
aa = -4.5681e+284
aa = 4.5681e+284
aa = -6.4603e+284
aa = 6.4603e+284
aa = -9.1362e+284
aa = 9.1362e+284
aa = -1.2921e+285
aa = 1.2921e+285
aa = -1.8272e+285
aa = 1.8272e+285
aa = -2.5841e+285
aa = 2.5841e+285
aa = -3.6545e+285
aa = 3.6545e+285
aa = -5.1682e+285
aa = 5.1682e+285
aa = -7.3090e+285
aa = 7.3090e+285
aa = -1.0336e+286
aa = 1.0336e+286
aa = -1.4618e+286
aa = 1.4618e+286
aa = -2.0673e+286
aa = 2.0673e+286
aa = -2.9236e+286
aa = 2.9236e+286
aa = -4.1346e+286
aa = 4.1346e+286
aa = -5.8472e+286
aa = 5.8472e+286
aa = -8.2692e+286
aa = 8.2692e+286
aa = -1.1694e+287
aa = 1.1694e+287
aa = -1.6538e+287
aa = 1.6538e+287
aa = -2.3389e+287
aa = 2.3389e+287
aa = -3.3077e+287
aa = 3.3077e+287
aa = -4.6778e+287
aa = 4.6778e+287
aa = -6.6153e+287
aa = 6.6153e+287
aa = -9.3555e+287
aa = 9.3555e+287
aa = -1.3231e+288
aa = 1.3231e+288
aa = -1.8711e+288
aa = 1.8711e+288
aa = -2.6461e+288
aa = 2.6461e+288
aa = -3.7422e+288
aa = 3.7422e+288
aa = -5.2923e+288
aa = 5.2923e+288
aa = -7.4844e+288
aa = 7.4844e+288
aa = -1.0585e+289
aa = 1.0585e+289
aa = -1.4969e+289
aa = 1.4969e+289
aa = -2.1169e+289
aa = 2.1169e+289
aa = -2.9938e+289
aa = 2.9938e+289
aa = -4.2338e+289
aa = 4.2338e+289
aa = -5.9875e+289
aa = 5.9875e+289
aa = -8.4676e+289
aa = 8.4676e+289
aa = -1.1975e+290
aa = 1.1975e+290
aa = -1.6935e+290
aa = 1.6935e+290
aa = -2.3950e+290
aa = 2.3950e+290
aa = -3.3871e+290
aa = 3.3871e+290
aa = -4.7900e+290
aa = 4.7900e+290
aa = -6.7741e+290
aa = 6.7741e+290
aa = -9.5800e+290
aa = 9.5800e+290
aa = -1.3548e+291
aa = 1.3548e+291
aa = -1.9160e+291
aa = 1.9160e+291
aa = -2.7096e+291
aa = 2.7096e+291
aa = -3.8320e+291
aa = 3.8320e+291
aa = -5.4193e+291
aa = 5.4193e+291
aa = -7.6640e+291
aa = 7.6640e+291
aa = -1.0839e+292
aa = 1.0839e+292
aa = -1.5328e+292
aa = 1.5328e+292
aa = -2.1677e+292
aa = 2.1677e+292
aa = -3.0656e+292
aa = 3.0656e+292
aa = -4.3354e+292
aa = 4.3354e+292
aa = -6.1312e+292
aa = 6.1312e+292
aa = -8.6709e+292
aa = 8.6709e+292
aa = -1.2262e+293
aa = 1.2262e+293
aa = -1.7342e+293
aa = 1.7342e+293
aa = -2.4525e+293
aa = 2.4525e+293
aa = -3.4683e+293
aa = 3.4683e+293
aa = -4.9050e+293
aa = 4.9050e+293
aa = -6.9367e+293
aa = 6.9367e+293
aa = -9.8100e+293
aa = 9.8100e+293
aa = -1.3873e+294
aa = 1.3873e+294
aa = -1.9620e+294
aa = 1.9620e+294
aa = -2.7747e+294
aa = 2.7747e+294
aa = -3.9240e+294
aa = 3.9240e+294
aa = -5.5493e+294
aa = 5.5493e+294
aa = -7.8480e+294
aa = 7.8480e+294
aa = -1.1099e+295
aa = 1.1099e+295
aa = -1.5696e+295
aa = 1.5696e+295
aa = -2.2197e+295
aa = 2.2197e+295
aa = -3.1392e+295
aa = 3.1392e+295
aa = -4.4395e+295
aa = 4.4395e+295
aa = -6.2784e+295
aa = 6.2784e+295
aa = -8.8790e+295
aa = 8.8790e+295
aa = -1.2557e+296
aa = 1.2557e+296
aa = -1.7758e+296
aa = 1.7758e+296
aa = -2.5113e+296
aa = 2.5113e+296
aa = -3.5516e+296
aa = 3.5516e+296
aa = -5.0227e+296
aa = 5.0227e+296
aa = -7.1032e+296
aa = 7.1032e+296
aa = -1.0045e+297
aa = 1.0045e+297
aa = -1.4206e+297
aa = 1.4206e+297
aa = -2.0091e+297
aa = 2.0091e+297
aa = -2.8413e+297
aa = 2.8413e+297
aa = -4.0182e+297
aa = 4.0182e+297
aa = -5.6825e+297
aa = 5.6825e+297
aa = -8.0363e+297
aa = 8.0363e+297
aa = -1.1365e+298
aa = 1.1365e+298
aa = -1.6073e+298
aa = 1.6073e+298
aa = -2.2730e+298
aa = 2.2730e+298
aa = -3.2145e+298
aa = 3.2145e+298
aa = -4.5460e+298
aa = 4.5460e+298
aa = -6.4291e+298
aa = 6.4291e+298
aa = -9.0921e+298
aa = 9.0921e+298
aa = -1.2858e+299
aa = 1.2858e+299
aa = -1.8184e+299
aa = 1.8184e+299
aa = -2.5716e+299
aa = 2.5716e+299
aa = -3.6368e+299
aa = 3.6368e+299
aa = -5.1432e+299
aa = 5.1432e+299
aa = -7.2736e+299
aa = 7.2736e+299
aa = -1.0286e+300
aa = 1.0286e+300
aa = -1.4547e+300
aa = 1.4547e+300
aa = -2.0573e+300
aa = 2.0573e+300
aa = -2.9095e+300
aa = 2.9095e+300
aa = -4.1146e+300
aa = 4.1146e+300
aa = -5.8189e+300
aa = 5.8189e+300
aa = -8.2292e+300
aa = 8.2292e+300
aa = -1.1638e+301
aa = 1.1638e+301
aa = -1.6458e+301
aa = 1.6458e+301
aa = -2.3276e+301
aa = 2.3276e+301
aa = -3.2917e+301
aa = 3.2917e+301
aa = -4.6551e+301
aa = 4.6551e+301
aa = -6.5833e+301
aa = 6.5833e+301
aa = -9.3103e+301
aa = 9.3103e+301
aa = -1.3167e+302
aa = 1.3167e+302
aa = -1.8621e+302
aa = 1.8621e+302
aa = -2.6333e+302
aa = 2.6333e+302
aa = -3.7241e+302
aa = 3.7241e+302
aa = -5.2667e+302
aa = 5.2667e+302
aa = -7.4482e+302
aa = 7.4482e+302
aa = -1.0533e+303
aa = 1.0533e+303
aa = -1.4896e+303
aa = 1.4896e+303
aa = -2.1067e+303
aa = 2.1067e+303
aa = -2.9793e+303
aa = 2.9793e+303
aa = -4.2133e+303
aa = 4.2133e+303
aa = -5.9586e+303
aa = 5.9586e+303
aa = -8.4267e+303
aa = 8.4267e+303
aa = -1.1917e+304
aa = 1.1917e+304
aa = -1.6853e+304
aa = 1.6853e+304
aa = -2.3834e+304
aa = 2.3834e+304
aa = -3.3707e+304
aa = 3.3707e+304
aa = -4.7669e+304
aa = 4.7669e+304
aa = -6.7413e+304
aa = 6.7413e+304
aa = -9.5337e+304
aa = 9.5337e+304
aa = -1.3483e+305
aa = 1.3483e+305
aa = -1.9067e+305
aa = 1.9067e+305
aa = -2.6965e+305
aa = 2.6965e+305
aa = -3.8135e+305
aa = 3.8135e+305
aa = -5.3931e+305
aa = 5.3931e+305
aa = -7.6270e+305
aa = 7.6270e+305
aa = -1.0786e+306
aa = 1.0786e+306
aa = -1.5254e+306
aa = 1.5254e+306
aa = -2.1572e+306
aa = 2.1572e+306
aa = -3.0508e+306
aa = 3.0508e+306
aa = -4.3145e+306
aa = 4.3145e+306
aa = -6.1016e+306
aa = 6.1016e+306
aa = -8.6289e+306
aa = 8.6289e+306
aa = -1.2203e+307
aa = 1.2203e+307
aa = -1.7258e+307
aa = 1.7258e+307
aa = -2.4406e+307
aa = 2.4406e+307
aa = -3.4516e+307
aa = 3.4516e+307
aa = -4.8813e+307
aa = 4.8813e+307
aa = -6.9031e+307
aa = 6.9031e+307
aa = -9.7625e+307
aa = 9.7625e+307
aa = -1.3806e+308
aa = 1.3806e+308
aa = -Inf
Exiting fzero: aborting search for an interval containing a sign change
because NaN or Inf function value encountered during search.
(Function value at -Inf is NaN.)
Check function or try again with a different starting value.
aa = NaN
function res = fun(aa,dd,vv,L)
aa
res = ( (-(sqrt(L^2-(vv)^2)/2)) + (aa*sinh(((dd)/2)/aa)) );
end
VIGNESH BALAJI
2024-2-7
I used the same code. If you see your output. One of the output was aa = 0.0142 (iteration number 22 in your current output of aa and iteration 23 in your previous output) which was the required answer.
I did a further analysis in the code and found the value of the function at aa = 0.0142 for the code and I got the answer as -0.0012 which means the real answer is very closeby in some digits, maybe a change in some decimal digits.
I have added the modified code below
dd = 0.1069;
vv = 0.7889;
L = 1;
xx0 = 0.15;
options = optimset('Display','iter'); % show iterations
[aa, fval, exitflag] = fzero(@(x)fun(x,dd,vv,L),xx0, options);
function res = fun(aa,dd,vv,L)
% aa
res = ( (-(sqrt(L^2-(vv)^2)/2)) + (aa*sinh(((dd)/2)/aa)) );
end
If I ran this code, you can see that at iteration 23 the value was very close to zero
Search for an interval around 0.15 containing a sign change:
Func-count a f(a) b f(b) Procedure
1 0.15 -0.252672 0.15 -0.252672 initial interval
3 0.145757 -0.252605 0.154243 -0.252735 search
5 0.144 -0.252575 0.156 -0.252759 search
7 0.141515 -0.252531 0.158485 -0.252792 search
9 0.138 -0.252464 0.162 -0.252836 search
11 0.133029 -0.252361 0.166971 -0.252893 search
13 0.126 -0.252193 0.174 -0.252966 search
15 0.116059 -0.251901 0.183941 -0.253055 search
17 0.102 -0.251331 0.198 -0.253159 search
19 0.0821177 -0.249956 0.217882 -0.253273 search
21 0.054 -0.244645 0.246 -0.253389 search
23 0.0142355 -0.00334771 0.285765 -0.253499 search
25 -0.042 -0.238169 0.342 -0.253593 search
Then it diverged without searching that region finer. I am guessing it is skipping the search region. I am unaware of what to do to make it search properly. It is clear that the gradient descent (I don't know if gradient descent is used in fzero/ bisection method) is showing a path to the answer and after that it diverges.
The actual exact answer of the problem is 0.0141792 which is very close to 0.0142355.
Please let me know your suggestions on this.
Torsten
2024-2-7
If you look at the f(a) and f(b) values in your output, you can see that both are negative in all iterations. Thus fzero does not find a sign change and keeps on serching for suitable limits a and b for which f(a)*f(b) < 0 holds.
VIGNESH BALAJI
2024-2-9
@Torsten thanks, I now understand the reason behidn fzero error. Both your answer and @Dyuman Joshi heped me to solve this problem. I am not sure how to accept both of your answers to close this thread.
回答(2 个)
Matt J
2024-2-2
编辑:Matt J
2024-2-2
You should plot the function to get a better idea where the solution is,
dd = 0.1069;
vv = 0.7889;
L = 1;
fun = @(aa) ( (-(sqrt(L^2-(vv)^2)/2)) + (aa*sinh(((dd)/2)/aa)) );
fplot(fun,[0.01,0.05])
Warning: Function behaves unexpectedly on array inputs. To improve performance, properly vectorize your function to return an output with the same size and shape as the input arguments.
fzero(fun,[0.01,0.05])
ans = 0.0142
2 个评论
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