How to find nearest next/previous double value (numeric_limits::epsilon for given number)

Question

The title is quite self-explanatory, input is given double value, and I want to add/substract the smallest amount possible.

Answer 1

You can use nextafter, which is available if your compiler implements C99's math functions (i.e., C++11 and above). This function (and its various overloads) can be described as:

double nextafter(double value, double target);

It will move from value in the direction of target by the smallest possible amount (typically by tweaking the bit representation of the float). If value is already at target, this does nothing.

If target is greater than value this will increment value by the smallest possible amount. If target is less than value this will decrement value by the smallest possible amount.

Common usage is to pass either DBL_MAX or INFINITY as the target to increase the minimal amount (or, the negation of them to decrease the minimal amount).

The choice between DBL_MAX and INFINITY depends on what you want to do at the boundary - nextafter(DBL_MAX, DBL_MAX) == DBL_MAX, but nextafter(DBL_MAX, INFINITY) == INFINITY.

And yes, it's poorly named. See also nextafter and nexttoward: why this particular interface?.

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#include <cfloat> // DBL_MAX
#include <cmath> // std::nextafter

double x = 0.1;

// Next representable number **after** x in the direction of DBL_MAX.
// (i.e., this is larger than x, and there is no double between it and x)
double xPlusSmallest = std::nextafter(x, DBL_MAX); 

// Next representable number **before** x in the direction of -DBL_MAX.
// (i.e., this is smaller than x, and there is no double between it and x)
double xMinusSmallest = std::nextafter(x, -DBL_MAX); 

Even if your compiler doesn't support it, it probably has an instrinsic for it. (MSVC has had _nextafter since 2005, for example. GCC probably implements it as standard.)

If your compiler doesn't support it but Boost is available to you, you can do this:

#include <boost/math/special_functions/next.hpp>
#include <cfloat> 

double x = 0.1;

double xPlusSmallest = boost::math::nextafter(x, DBL_MAX); 
double xMinusSmallest = boost::math::nextafter(x, -DBL_MAX); 

And if none of those work for you, you'll just have to crack open the Boost header and copy it.

Answer 2

Here's a very dirty trick that isn't actually legal and only works if your platform uses IEEE754 floats: The binary representation of the float is ordered in the same way as the float value, so you can increment the binary representation:

double x = 1.25;

uint64_t * const p = reinterpret_cast<uint64_t*>(&x);

++*p;   // undefined behaviour! but it gets the next value

// now x has the next value

You can achieve the same effect entirely legally by doing the usual binary copy gymnastics to obtain a proper uint64_t value. Make sure to check for zero, infinity and NaN properly, too.