In[]:=
c=NIntegrate[E^(-t^2),{t,0,1/2},WorkingPrecision->20]
Out[]=
0.46128100641279244876
By https://mathworld.wolfram.com/CubeLinePicking.html,
let r be the average distance between two points chosen at random inside a unit cube:
In[]:=
r=PolyhedronData["Cube","MeanInteriorLineSegmentLength"];
In[]:=
n=
3(7/5r-2)
2(26847/5r-5001)
;
In[]:=
Quiet
1
c+n
(c+n)

Out[]=
0.18785964246206712290

In[]:=
0.18785964246206712290
Input interpretation
0.18785964246206712290
Number line
Rational form
1878596424620671229
10000000000000000000
Number name
zero point one eight seven eight five nine six four two four six two zero six seven
Continued fraction
Fraction form
[0; 5, 3, 10, 1, 1, 4, 1, 1, 1, 1, 9, 1, 1, 12, 2, 17, 2, 3, 2, 1, 1, 1, 2]
Possible closed forms
More

MRB
≈0.1878596424620671202485
root of 47663
3
x
+76535
2
x
-21234x+972 near x0.18786
≈0.18785964246206712289677
-tancot
11292147
6436471
≈0.18785964246206712251983
C_MRB is the MRB constant »
cot(x) is the cotangent function »
So define m as
∞
∫
1

rt

csc(πt)t
where
r=
log(
1/t
t
-1)
t
:
In[]:=
m=NReNIntegrate(
rt

Csc[πt])/.r->Log[t^(1/t)-1]t,{t,1,I*Infinity},WorkingPrecision->20,Method->"Trapezoidal",18
Out[]=
0.187859642462067120
In[]:=
Quietc-
-nLog[m]-ProductLog[-Log[m]]
Log[m]

Out[]=
2.×
-18
10
So,c≈
-nlog(m)-W(-log(m))
log(m)
,wheremistheMRBconstantandnisatermoftheInteriorLineSegmentLength.