log1p

Function: log1p
Section: transcendental
C-Name: glog1p
Prototype: Gp
Help: log1p(x): log(1+x)
Doc: return $\log(1+x)$, computed in a way that is also accurate
 when the real part of $x$ is near $0$. This is the reciprocal function
 of \kbd{expm1}$(x) = \exp(x)-1$.
 \bprog
 ? default(realprecision, 10000); x = Pi*1e-100;
 ? (expm1(log1p(x)) - x) / x
 %2 = -7.668242895059371866 E-10019
 ? (log1p(expm1(x)) - x) / x
 %3 = -7.668242895059371866 E-10019
 @eprog\noindent When $x$ is small, this function is both faster and more
 accurate than $\log(1+x)$:
 \bprog
 ? \p38
 ? x = 1e-20;
 ? localprec(100); c = log1p(x); \\ reference point
 ? a = log1p(x); abs((a - c)/c)
 %6 = 0.E-38
 ? b = log(1+x); abs((b - c)/c)  \\ slightly less accurate
 %7 = 1.5930919111324522770 E-38
 ? for (i=1,10^5,log1p(x))
 time = 81 ms.
 ? for (i=1,10^5,log(1+x))
 time = 100 ms. \\ slower, too
 @eprog