The answer is the same as Rita's. The method a little less cumbersome. It increases your significant figures by about 16 as compared with simply subtracting.
123456789^2 = 15241578750190521, so you want to find f(123456789) when f(x) = x - sqrt(x^2-1). Most calculators would have a problem with this, however any CAS eg Maple can do it effortlessly: the answer is
0.0000000040500000368550004018106298655
to considerably more than 20 figures.
Presumably, there is some way to do it with Taylor series, or other numerical method, but having maple handy discourages thinking about that.
Answers & Comments
Verified answer
Looking at Rita's work, the way to calculate it is:
x^2 -y^2 = 1 = (x-y)(x+y)
x-y = 1/(x+y)
x = 123456789
y = √15241578750190520
x+y = 246913577.999999995949999963145 ...
x-y = 1/(x+y) = 4.050 000 036 855 000 401 810 629 865 505 e-9
The answer is the same as Rita's. The method a little less cumbersome. It increases your significant figures by about 16 as compared with simply subtracting.
123456789^2 = 15241578750190521, so you want to find f(123456789) when f(x) = x - sqrt(x^2-1). Most calculators would have a problem with this, however any CAS eg Maple can do it effortlessly: the answer is
0.0000000040500000368550004018106298655
to considerably more than 20 figures.
Presumably, there is some way to do it with Taylor series, or other numerical method, but having maple handy discourages thinking about that.
I see no reason to take this question seriously.