/usr/share/openturns/validation/Cas4.txt is in openturns-validation 1.5-7build2.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
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> Digits:=20:
> G:=1/2*(x1-x2)^2+(x1+x2)/sqrt(2)-3:
> with(plots):
> implicitplot(G,x1=-5..5,x2=-5..5);
Warning, the name changecoords has been redefined
>
> solve(G,x2);
> Pdef:=1-int(int(1/(2*Pi)*exp(-(x1^2+x2^2)/2),x2=-1/2*2^(1/2)+x1-1/2*(2
> 6-8*2^(1/2)*x1)^(1/2)..-1/2*2^(1/2)+x1+1/2*(26-8*2^(1/2)*x1)^(1/2)),x1
> =-infinity..26/(8*sqrt(2)));
1/2 1/2 1/2
2 (26 - 8 2 x1)
- ---- + x1 + -------------------,
2 2
1/2 1/2 1/2
2 (26 - 8 2 x1)
- ---- + x1 - -------------------
2 2
1/2
13 2
-------
8
/ 2 /
| x1 1/2 |
Pdef := 1 - | 1/4 exp(- ---) 2 |
| 2 \
/
-infinity
1/2 1/2 1/2 1/2
2 x1 2 (26 - 8 2 x1)
erf(- 1/2 + ------- + ------------------------)
2 4
1/2 1/2 1/2 1/2 \
2 x1 2 (26 - 8 2 x1) | / 1/2
- erf(- 1/2 + ------- - ------------------------)| / Pi
2 4 / /
dx1
> Pdef_num:=evalf(Pdef);
Pdef_num := 0.10456369317559775009
> PHI:=proc(x)
> int(1/sqrt(2*Pi)*exp(-t^2/2),t=-infinity..x)
> end:
> -fsolve(PHI(x)=Pdef_num,x);
1.2559685128742776753
>
> 1-7.1529e-3/9.81929e-3;
0.27154610974927922487
>
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