...one of the most highly
regarded and expertly designed C++ library projects in the
world.

— Herb Sutter and Andrei
Alexandrescu, C++
Coding Standards

#include <boost/math/special_functions/log1p.hpp>

namespace boost{ namespace math{ template <class T>calculated-result-typelog1p(T x); template <class T, class Policy>calculated-result-typelog1p(T x, const Policy&); }} // namespaces

Returns the natural logarithm of `x+1`

.

The return type of this function is computed using the *result
type calculation rules*: the return is `double`

when *x* is an integer type and T otherwise.

The final Policy argument is optional and can be used to control the behaviour of the function: how it handles errors, what level of precision to use etc. Refer to the policy documentation for more details.

There are many situations where it is desirable to compute `log(x+1)`

.
However, for small `x`

then
`x+1`

suffers from catastrophic cancellation errors
so that `x+1 == 1`

and `log(x+1) == 0`

,
when in fact for very small x, the best approximation to `log(x+1)`

would be
`x`

. `log1p`

calculates the best approximation to `log(1+x)`

using
a Taylor series expansion for accuracy (less than 2ɛ). Alternatively note that
there are faster methods available, for example using the equivalence:

log(1+x) == (log(1+x) * x) / ((1+x) - 1)

However, experience has shown that these methods tend to fail quite spectacularly once the compiler's optimizations are turned on, consequently they are used only when known not to break with a particular compiler. In contrast, the series expansion method seems to be reasonably immune to optimizer-induced errors.

Finally when BOOST_HAS_LOG1P is defined then the `float/double/long double`

specializations of this template simply forward to the platform's native
(POSIX) implementation of this function.

The following graph illustrates the behaviour of log1p:

For built in floating point types `log1p`

should have approximately 1 epsilon accuracy.

A mixture of spot test sanity checks, and random high precision test values calculated using NTL::RR at 1000-bit precision.