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For $b$ not much larger than $a$, the maximum difference between $e^x$ and the line from $(a,e^a)$ to $(b,e^b)$, for $x\in[a,b]$, is approximately $\frac 18 (b-a)^2 e^a$. So you should choose your poi …

Modern mathematics libraries should be able to find $\log x$ precisely for all floating-point numbers, as the algorithms for doing that have long been known and adopted. My experiments on a fairly rec …

As Pietro says, $u=\exp(W(x))$ satisfies $u=x/\ln u$. This is a contraction mapping for large enough $u$, so just start with any old approximation, like $u=x/\ln x$, and do $u:=x/\ln u$ until it conv …

If $Y(x)=(1-\Phi(x))/\phi(x)$, it is easy to check that $Y'(x)=xY(x)-1$ and from this anything you like follows by standard methods.

Here is one approach you can use. The differences $a_k$ provide you with bounds on the difference between the characteristic functions of the two distributions. This may be easy or not depending on …

I'll give a crude calculation on the back of this envelope. Assume that $c'm<r<cm$ for some $0<c'<c<\frac12$. When $i=o(r^{1/2})$, we have $$\binom{m}{r-i}= (1+o(1))\, \left(\frac{r}{m-r}\right)^i\bin …

Version 2 Here is some experimental evidence. Like most people here I'm using $-f(x)$ and assuming $0<b<a<1$. Consider $b=1/100$ and $a=i/100$. For $2\le i\le 3$, there is a quadratic polynomial. F …

My paper here (Adv. Appl. Prob., 21 (1989) 475-478), Theorem 2, provides an estimate over all values of the parameters with relative error that is $o(1)$ if either $\sigma\to\infty$ or $x\sigma\to\inf …