> restart:
> a:=-1:
> b:=1:
> PhiA:=int(1/sqrt(2*Pi)*exp(-x^2/2),x=-infinity..a):
> evalf[16](PhiA);
> PhiB:=int(1/sqrt(2*Pi)*exp(-x^2/2),x=-infinity..b);
> evalf[16](PhiB);
> phiA:=1/sqrt(2*Pi)*exp(-a^2/2);
> evalf[16](phiA);
> phiB:=1/sqrt(2*Pi)*exp(-b^2/2);
> evalf[16](phiB);
> normalizationFactor:=1/(PhiB - PhiA);
> evalf[16](normalizationFactor);
> phiANorm:=phiA*normalizationFactor;
> evalf[16](phiANorm);
> phiBNorm:=phiB*normalizationFactor;
> evalf[16](phiBNorm);
> PhiANorm:=PhiA*normalizationFactor;
> evalf[16](PhiANorm);
> PhiBNorm:=PhiB*normalizationFactor;
> evalf[16](PhiBNorm);

                          0.1586552539314570


                        / 1/2   1/2                       1/2 \
                    1/2 |2    Pi           1/2   1/2     2    |
                   2    |---------- + 1/2 2    Pi    erf(----)|
                        \    2                            2   /
       PhiB := 1/2 --------------------------------------------
                                        1/2
                                      Pi


                          0.8413447460685425


                                   1/2
                                  2    exp(-1/2)
                      phiA := 1/2 --------------
                                        1/2
                                      Pi


                          0.2419707245191433


                                   1/2
                                  2    exp(-1/2)
                      phiB := 1/2 --------------
                                        1/2
                                      Pi


                          0.2419707245191433


                           /
                           |
                           |
                           |
  normalizationFactor := 1/|
                           |
                           \

                 / 1/2   1/2                       1/2 \
             1/2 |2    Pi           1/2   1/2     2    |
            2    |---------- + 1/2 2    Pi    erf(----)|
                 \    2                            2   /
        1/2 --------------------------------------------
                                 1/2
                               Pi

                    / 1/2   1/2                       1/2 \\
                1/2 |2    Pi           1/2   1/2     2    ||
               2    |---------- - 1/2 2    Pi    erf(----)||
                    \    2                            2   /|
         - 1/2 --------------------------------------------|
                                    1/2                    |
                                  Pi                       /


                          1.464794773491545


                                     /      /
                                     |      |
                                     |      |
                   1/2             / |  1/2 |
  phiANorm := 1/2 2    exp(-1/2)  /  |Pi    |
                                 /   |      |
                                     \      \

                 / 1/2   1/2                       1/2 \
             1/2 |2    Pi           1/2   1/2     2    |
            2    |---------- + 1/2 2    Pi    erf(----)|
                 \    2                            2   /
        1/2 --------------------------------------------
                                 1/2
                               Pi

                    / 1/2   1/2                       1/2 \\\
                1/2 |2    Pi           1/2   1/2     2    |||
               2    |---------- - 1/2 2    Pi    erf(----)|||
                    \    2                            2   /||
         - 1/2 --------------------------------------------||
                                    1/2                    ||
                                  Pi                       //


                          0.3544374526136036


                                     /      /
                                     |      |
                                     |      |
                   1/2             / |  1/2 |
  phiBNorm := 1/2 2    exp(-1/2)  /  |Pi    |
                                 /   |      |
                                     \      \

                 / 1/2   1/2                       1/2 \
             1/2 |2    Pi           1/2   1/2     2    |
            2    |---------- + 1/2 2    Pi    erf(----)|
                 \    2                            2   /
        1/2 --------------------------------------------
                                 1/2
                               Pi

                    / 1/2   1/2                       1/2 \\\
                1/2 |2    Pi           1/2   1/2     2    |||
               2    |---------- - 1/2 2    Pi    erf(----)|||
                    \    2                            2   /||
         - 1/2 --------------------------------------------||
                                    1/2                    ||
                                  Pi                       //


                          0.3544374526136036


                                                                   /
                                                                   |
                       / 1/2   1/2                       1/2 \     |
                   1/2 |2    Pi           1/2   1/2     2    |   / |
  PhiANorm := 1/2 2    |---------- - 1/2 2    Pi    erf(----)|  /  |
                       \    2                            2   / /   |
                                                                   \

              /         / 1/2   1/2                       1/2 \
              |     1/2 |2    Pi           1/2   1/2     2    |
              |    2    |---------- + 1/2 2    Pi    erf(----)|
          1/2 |         \    2                            2   /
        Pi    |1/2 --------------------------------------------
              |                         1/2
              \                       Pi

                    / 1/2   1/2                       1/2 \\\
                1/2 |2    Pi           1/2   1/2     2    |||
               2    |---------- - 1/2 2    Pi    erf(----)|||
                    \    2                            2   /||
         - 1/2 --------------------------------------------||
                                    1/2                    ||
                                  Pi                       //


                          0.2323973867457722


                                                                   /
                                                                   |
                       / 1/2   1/2                       1/2 \     |
                   1/2 |2    Pi           1/2   1/2     2    |   / |
  PhiBNorm := 1/2 2    |---------- + 1/2 2    Pi    erf(----)|  /  |
                       \    2                            2   / /   |
                                                                   \

              /         / 1/2   1/2                       1/2 \
              |     1/2 |2    Pi           1/2   1/2     2    |
              |    2    |---------- + 1/2 2    Pi    erf(----)|
          1/2 |         \    2                            2   /
        Pi    |1/2 --------------------------------------------
              |                         1/2
              \                       Pi

                    / 1/2   1/2                       1/2 \\\
                1/2 |2    Pi           1/2   1/2     2    |||
               2    |---------- - 1/2 2    Pi    erf(----)|||
                    \    2                            2   /||
         - 1/2 --------------------------------------------||
                                    1/2                    ||
                                  Pi                       //


                          1.232397386745772

> evalf[20](1/(int(1/sqrt(2*Pi)*exp(-x^2/2),x=-infinity..1)-int(1/sqrt(2
> *Pi)*exp(-x^2/2),x=-infinity..-1)));

                        1.4647947734915440800

> evalf[20](1/(sqrt(3-2)*Beta(1/2,3/2)));

                        0.63661977236758134306

> evalf[20](sqrt(1/3));

                        0.57735026918962576449

> evalf[20](1/sqrt(2*Pi));

                        0.39894228040143267794

> 
