[section:next_float Floating-Point Representation Distance (ULP), and Finding Adjacent Floating-Point Values] [@http://en.wikipedia.org/wiki/Unit_in_the_last_place Unit of Least Precision or Unit in the Last Place] is the gap between two different, but as close as possible, floating-point numbers. Most decimal values, for example 0.1, cannot be exactly represented as floating-point values, but will be stored as the [@http://en.wikipedia.org/wiki/Floating_point#Representable_numbers.2C_conversion_and_rounding closest representable floating-point]. Functions are provided for finding adjacent greater and lesser floating-point values, and estimating the number of gaps between any two floating-point values. The floating-point type FPT must have has a fixed number of bits in the representation. The number of bits may set at runtime, but must be the same for all numbers. For example, __NTL_quad_float type (fixed 128-bit representation) or __NTL_RR type (arbitrary but fixed decimal digits, default 150) but *not* a type that extends the representation to provide an exact representation for any number, for example [@http://keithbriggs.info/xrc.html XRC eXact Real in C]. [section:nextafter Finding the Next Representable Value in a Specific Direction (nextafter)] [h4 Synopsis] `` #include `` namespace boost{ namespace math{ template FPT nextafter(FPT val, FPT direction); }} // namespaces [h4 Description - nextafter] This is an implementation of the `nextafter` function included in the C99 standard. (It is also effectively an implementation of the C99 'nexttoward' legacy function which differs only having a long double direction, and can generally serve in its place if required). [note The C99 functions must use suffixes f and l to distinguish float and long double versions. C++ uses the template mechanism instead.] Returns the next representable value after /x/ in the direction of /y/. If `x == y` then returns /x/. If /x/ is non-finite then returns the result of a __domain_error. If there is no such value in the direction of /y/ then returns an __overflow_error. [warning The template parameter FTP must be a floating-point type. An integer type, for example, will produce an unhelpful error message.] [tip Nearly always, you just want the next or prior representable value, so instead use `float_next` or `float_prior` below.] [h4 Examples - nextafter] The two representations using a 32-bit float either side of unity are: `` The nearest (exact) representation of 1.F is 1.00000000 nextafter(1.F, 999) is 1.00000012 nextafter(1/f, -999) is 0.99999994 The nearest (not exact) representation of 0.1F is 0.100000001 nextafter(0.1F, 10) is 0.100000009 nextafter(0.1F, 10) is 0.099999994 `` [endsect] [section:float_next Finding the Next Greater Representable Value (float_next)] [h4 Synopsis] `` #include `` namespace boost{ namespace math{ template FPT float_next(FPT val); }} // namespaces [h4 Description - float_next] Returns the next representable value which is greater than /x/. If /x/ is non-finite then returns the result of a __domain_error. If there is no such value greater than /x/ then returns an __overflow_error. Has the same effect as nextafter(val, (std::numeric_limits::max)()); [endsect] [/section:float_next Finding the Next Greater Representable Value (float_prior)] [section:float_prior Finding the Next Smaller Representable Value (float_prior)] [h4 Synopsis] `` #include `` namespace boost{ namespace math{ template FPT float_prior(FPT val); }} // namespaces [h4 Description - float_prior] Returns the next representable value which is less than /x/. If /x/ is non-finite then returns the result of a __domain_error. If there is no such value less than /x/ then returns an __overflow_error. Has the same effect as nextafter(val, -(std::numeric_limits::max)()); // Note most negative value -max. [endsect] [/section:float_prior Finding the Next Smaller Representable Value (float_prior)] [section:float_distance Calculating the Representation Distance Between Two Floating Point Values (ULP) float_distance] Function float_distance finds the number of gaps/bits/ULP between any two floating-point values. If the significands of floating-point numbers are viewed as integers, then their difference is the number of ULP/gaps/bits different. [h4 Synopsis] `` #include `` namespace boost{ namespace math{ template FPT float_distance(FPT a, FPT b); }} // namespaces [h4 Description - float_distance] Returns the distance between /a/ and /b/: the result is always a signed integer value (stored in floating-point type FPT) representing the number of distinct representations between /a/ and /b/. Note that * `float_distance(a, a)` always returns 0. * `float_distance(float_next(a), a)` always returns -1. * `float_distance(float_prior(a), a)` always returns 1. The function `float_distance` is equivalent to calculating the number of ULP (Units in the Last Place) between /a/ and /b/ except that it returns a signed value indicating whether `a > b` or not. If the distance is too great then it may not be able to be represented as an exact integer by type FPT, but in practice this is unlikely to be a issue. [endsect] [/section:float_distance Calculating the Representation Distance Between Two Floating Point Values (ULP) float_distance] [section:float_advance Advancing a Floating Point Value by a Specific Representation Distance (ULP) float_advance] Function float_advance advances a floating point number by a specified number of ULP. [h4 Synopsis] `` #include `` namespace boost{ namespace math{ template FPT float_advance(FPT val, int distance); }} // namespaces [h4 Description - float_advance] Returns a floating point number /r/ such that `float_distance(val, r) == distance`. [endsect] [/section:float_advance] [endsect] [/ section:next_float Floating-Point Representation Distance (ULP), and Finding Adjacent Floating-Point Values] [/ Copyright 2008 John Maddock and Paul A. Bristow. Distributed under the Boost Software License, Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt). ]