...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> calculatedresulttype log1p(T x); template <class T, class Policy> calculatedresulttype log1p(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 optimizerinduced 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.
Table 7.81. Error rates for log1p
GNU C++ version 7.1.0 
GNU C++ version 7.1.0 
Sun compiler version 0x5150 
Microsoft Visual C++ version 14.1 


Random test data 
Max = 0.818ε (Mean = 0.227ε) 
Max = 0.846ε (Mean = 0.153ε) 
Max = 2.3ε (Mean = 0.66ε) 
Max = 0.509ε (Mean = 0.057ε) 
A mixture of spot test sanity checks, and random high precision test values calculated using NTL::RR at 1000bit precision.