Originally Posted by: manhtruong Very helpful, thank you very much!
Welcome ... read more of interest. So savant a computer may be,
NONE contains mathematical functions, only the 4 arithmetic
operations [+,-,*,/] ... logic, memory as well.
Common functions: Exp(x), ln(x), X^Y [Power function], trig, SQRT...
come with Windows. More advanced functions [Si, Ei, Bessel ...]
are implemented by the software you use [Mathematica, Maple, ...]
In PC's, common functions are approximated from normalised
rational fractions giving 18/21 decimals, from which only 15 are
conserved and displayed. Main frames have supplementary base accuracy
from Chebyshev 25 decimals very stable.
As far as now, Smath will not display > 15 decimals [like Excel...].
Mathcad, Maple, Mathematica ... run what they call "Big Engine"
that will deploy all the floating point register. At the beginning
of the "Main Frames" approximation of functions were done from
Continued Fractions, slow but infinitely unlimited. Thus, Mathcad
could output 250/400 decimals for sin(something), SQRT(2) ... etc.
References:
1. C.T. Fike [Continued Fractions]
2. Luke, Clenshaw [Chebysev]
3. Cody junior [general implementation of rational fractions]
4. Hart et Al. [all collected rational fractions]
Have a good day and enjoy Smath ... Jean