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#include <boost/math/special_functions/sqrt1pm1.hpp>

namespace boost{ namespace math{

template <class T>
calculated-result-type sqrt1pm1(T x);

template <class T, class Policy>
calculated-result-type sqrt1pm1(T x, const Policy&);

}} // namespaces

Returns sqrt(1+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.

This function is useful when you need the difference between sqrt(x) and 1, when x is itself close to 1.

Implemented in terms of log1p and expm1.

The following graph illustrates the behaviour of sqrt1pm1:


For built in floating-point types sqrt1pm1 should have approximately 3 epsilon accuracy.

Table 8.84. Error rates for sqrt1pm1

GNU C++ version 7.1.0

GNU C++ version 7.1.0
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1


Max = 1.3ε (Mean = 0.404ε)

Max = 1.33ε (Mean = 0.404ε)

Max = 1.54ε (Mean = 0.563ε)

Max = 1.35ε (Mean = 0.497ε)


A selection of random high precision test values calculated using NTL::RR at 1000-bit precision.