expm1

#include <boost/math/special_functions/expm1.hpp>

namespace boost{ namespace math{

template <class T>
calculated-result-type expm1(T x);

template <class T, class Policy>
calculated-result-type expm1(T x, const Policy&);

}} // namespaces

Returns e^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.

For small x, then e^x is very close to 1, as a result calculating e^x - 1 results in catastrophic cancellation errors when x is small. expm1 calculates e^x - 1 using rational approximations (for up to 128-bit long doubles), otherwise via a series expansion when x is small (giving an accuracy of less than 2ɛ).

Finally when BOOST_HAS_EXPM1 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 expm1:

Accuracy

For built in floating point types expm1 should have approximately 1 epsilon accuracy.

Table 6.79. Error rates for expm1

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
Random test data	Max = 0.996ε (Mean = 0.283ε) (<math.h>: Max = 1.31ε (Mean = 0.496ε))	Max = 0.793ε (Mean = 0.126ε) (Rmath 3.0.2: Max = 0.793ε (Mean = 0.126ε)) (Cephes: Max = 1.53ε (Mean = 0.535ε))	Max = 0.992ε (Mean = 0.402ε) (<tr1/cmath>: Max = 1.26e+19ε (Mean = 4.89e+18ε) And other failures.) (<math.h>: Max = 0.992ε (Mean = 0.402ε))	Max = 1.31ε (Mean = 0.406ε) (<math.h>: Max = 0.996ε (Mean = 0.426ε))

Testing

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