The computation of the exponential integral
E
i
(
x
)
Ei(x)
,
x
>
0
x > 0
, using rational Chebyshev approximations is discussed. The necessary approximations are presented in well-conditioned forms for the intervals
(
0
,
6
]
(0,6]
,
[
6
,
12
]
[6,12]
,
[
12
,
24
]
[12,24]
and
[
24
,
∞
)
[24,\infty )
. Maximal relative errors are as low as from
8
×
10
−
19
t
o
2
×
10
−
21
8 \times {10^{ - 19}}to2 \times {10^{ - 21}}
. In addition, the value of the zero of
E
i
(
x
)
Ei(x)
is presented to 30 decimal places.