Accuracy of Integral with epsilon

[Albert Chan]

(10-09-2021 08:37 PM)robve Wrote:  There is never a guarantee that a given function is constant by probing a few points. That should be obvious, no?

Tanh-Sinh (and other DE variants) does not see a constant function ... it see a bell-shaped curve.

CAS> 'int(1,x,-1,1)' (x = tanh(sinh(t)))

\(\int _{-\infty }^{+\infty }\mathrm{cosh}\left(t\right)\cdot (1-\left(\mathrm{tanh}\left(\mathrm{sinh}\left(t\right)\right)\right)^{2})\, dt\)

(10-09-2021 09:50 PM)robve Wrote:  You do realize that it is a fallacy to assume exp(x) and cosh(x)+sinh(x) equate numerically, in addition to algebraically?

It is interesting that my MAPM implementation of raw_exp() actually use above "identity"
Of course, |x| is limited to tiny value, below 1E-4, so catastrophic cancellation is not an issue.

sinh(x) converge with half as many terms, compared with exp(x)

exp(x) = x + x^2/2! + x^3/3! + ...
sinh(x) = x + x^3/3! + x^5/5! + ...