expm1

Function: expm1
Section: transcendental
C-Name: gexpm1
Prototype: Gp
Help: expm1(x): exp(x)-1.
Description:
 (real):real         mpexpm1($1)
Doc: return $\exp(x)-1$, computed in a way that is also accurate
 when the real part of $x$ is near $0$.
 A naive direct computation would suffer from catastrophic cancellation;
 PARI's direct computation of $\exp(x)$ alleviates this well known problem at
 the expense of computing $\exp(x)$ to a higher accuracy when $x$ is small.
 Using \kbd{expm1} is recommended instead:
 \bprog
 ? default(realprecision, 10000); x = 1e-100;
 ? a = expm1(x);
 time = 4 ms.
 ? b = exp(x)-1;
 time = 4 ms.
 ? default(realprecision, 10040); x = 1e-100;
 ? c = expm1(x);  \\ reference point
 ? abs(a-c)/c  \\ relative error in expm1(x)
 %7 = 1.4027986153764843997 E-10019
 ? abs(b-c)/c  \\ relative error in exp(x)-1
 %8 = 1.7907031188259675794 E-9919
 @eprog\noindent As the example above shows, when $x$ is near $0$,
 \kbd{expm1} is more accurate than \kbd{exp(x)-1}.