Synopsis

#include <boost/math/special_functions/pow1p.hpp>

namespace boost {
namespace math {

template <typename T1, typename T2, typename Policy>
BOOST_MATH_GPU_ENABLED tools::promote_args_t<T1, T2>
pow1p(const T1 x, const T2 y, const Policy& pol)

template <typename T1, typename T2>
BOOST_MATH_GPU_ENABLED tools::promote_args_t<T1, T2>
pow1p(const T1 x, const T2 y)

}} // namespaces

The function pow1p computes (1 + x)^y where x and y are real numbers. This function is particularly useful for scenarios where adding 1 to x before raising it to the power of y is required. It provides a more numerically stable and efficient way to perform this computation, especially for small values of x. When x is very small, directly computing (1 + x)^y using standard arithmetic operations can lead to a loss of precision due to floating-point arithmetic limitations. The pow1p function helps mitigate this issue by internally handling such cases more accurately.

The type T must provide an fma function, found either as std::fma or by argument dependent lookup.

Accuracy

In random testing against a 100 digit reference, the error is at most 3 ULP for float and double, including tiny x with huge y, where pow(1 + x, y) loses all precision. In the plots below, pow1p is shown in blue and pow(1 + x, y) in orange; red points are errors that exceed 100 ULP.

	Warning
	pow1p relies on error-free transforms and IEEE overflow behavior. Compiling with value-unsafe floating-point optimizations (-ffast-math, /fp:fast, or icpx's default -fp-model=fast) can make results for small x and large y completely wrong. With icpx, use -fp-model=precise.