What would be the most efficient way to compare two double or two float values?
Simply doing this is not correct:
bool CompareDoubles1 (double A, double B) { return A == B; } But something like:
bool CompareDoubles2 (double A, double B) { diff = A - B; return (diff < EPSILON) && (-diff < EPSILON); } Seems to waste processing.
Does anyone know a smarter float comparer?
Be extremely careful using any of the other suggestions. It all depends on context.
I have spent a long time tracing a bugs in a system that presumed a==b if |a-b|<epsilon. The underlying problems were:
The implicit presumption in an algorithm that if a==b and b==c then a==c.
Using the same epsilon for lines measured in inches and lines measured in mils (.001 inch). That is a==b but 1000a!=1000b. (This is why AlmostEqual2sComplement asks for the epsilon or max ULPS).
The use of the same epsilon for both the cosine of angles and the length of lines!
Using such a compare function to sort items in a collection. (In this case using the builtin C++ operator == for doubles produced correct results.)
Like I said: it all depends on context and the expected size of a and b.
BTW, std::numeric_limits<double>::epsilon() is the "machine epsilon". It is the difference between 1.0 and the next value representable by a double. I guess that it could be used in the compare function but only if the expected values are less than 1. (This is in response to @cdv's answer...)
Also, if you basically have int arithmetic in doubles (here we use doubles to hold int values in certain cases) your arithmetic will be correct. For example 4.0/2.0 will be the same as 1.0+1.0. This is as long as you do not do things that result in fractions (4.0/3.0) or do not go outside of the size of an int.
The comparison with an epsilon value is what most people do (even in game programming).
You should change your implementation a little though:
bool AreSame(double a, double b) { return fabs(a - b) < EPSILON; } Edit: Christer has added a stack of great info on this topic on a recent blog post. Enjoy.
I found that the Google C++ Testing Framework contains a nice cross-platform template-based implementation of AlmostEqual2sComplement which works on both doubles and floats. Given that it is released under the BSD license, using it in your own code should be no problem, as long as you retain the license. I extracted the below code from http://code.google.com/p/googletest/source/browse/trunk/include/gtest/internal/gtest-internal.h and added the license on top.
Be sure to #define GTEST_OS_WINDOWS to some value (or to change the code where it's used to something that fits your codebase - it's BSD licensed after all).
Usage example:
double left = // something double right = // something const FloatingPoint<double> lhs(left), rhs(right); if (lhs.AlmostEquals(rhs)) { //they're equal! } Here's the code:
// Copyright 2005, Google Inc. // All rights reserved. // // Redistribution and use in source and binary forms, with or without // modification, are permitted provided that the following conditions are // met: // // * Redistributions of source code must retain the above copyright // notice, this list of conditions and the following disclaimer. // * Redistributions in binary form must reproduce the above // copyright notice, this list of conditions and the following disclaimer // in the documentation and/or other materials provided with the // distribution. // * Neither the name of Google Inc. nor the names of its // contributors may be used to endorse or promote products derived from // this software without specific prior written permission. // // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS // "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT // LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR // A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT // OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, // SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT // LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, // DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY // THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT // (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE // OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. // // Authors: wan@google.com (Zhanyong Wan), eefacm@gmail.com (Sean Mcafee) // // The Google C++ Testing Framework (Google Test) // This template class serves as a compile-time function from size to // type. It maps a size in bytes to a primitive type with that // size. e.g. // // TypeWithSize<4>::UInt // // is typedef-ed to be unsigned int (unsigned integer made up of 4 // bytes). // // Such functionality should belong to STL, but I cannot find it // there. // // Google Test uses this class in the implementation of floating-point // comparison. // // For now it only handles UInt (unsigned int) as that's all Google Test // needs. Other types can be easily added in the future if need // arises. template <size_t size> class TypeWithSize { public: // This prevents the user from using TypeWithSize<N> with incorrect // values of N. typedef void UInt; }; // The specialization for size 4. template <> class TypeWithSize<4> { public: // unsigned int has size 4 in both gcc and MSVC. // // As base/basictypes.h doesn't compile on Windows, we cannot use // uint32, uint64, and etc here. typedef int Int; typedef unsigned int UInt; }; // The specialization for size 8. template <> class TypeWithSize<8> { public: #if GTEST_OS_WINDOWS typedef __int64 Int; typedef unsigned __int64 UInt; #else typedef long long Int; // NOLINT typedef unsigned long long UInt; // NOLINT #endif // GTEST_OS_WINDOWS }; // This template class represents an IEEE floating-point number // (either single-precision or double-precision, depending on the // template parameters). // // The purpose of this class is to do more sophisticated number // comparison. (Due to round-off error, etc, it's very unlikely that // two floating-points will be equal exactly. Hence a naive // comparison by the == operation often doesn't work.) // // Format of IEEE floating-point: // // The most-significant bit being the leftmost, an IEEE // floating-point looks like // // sign_bit exponent_bits fraction_bits // // Here, sign_bit is a single bit that designates the sign of the // number. // // For float, there are 8 exponent bits and 23 fraction bits. // // For double, there are 11 exponent bits and 52 fraction bits. // // More details can be found at // http://en.wikipedia.org/wiki/IEEE_floating-point_standard. // // Template parameter: // // RawType: the raw floating-point type (either float or double) template <typename RawType> class FloatingPoint { public: // Defines the unsigned integer type that has the same size as the // floating point number. typedef typename TypeWithSize<sizeof(RawType)>::UInt Bits; // Constants. // # of bits in a number. static const size_t kBitCount = 8*sizeof(RawType); // # of fraction bits in a number. static const size_t kFractionBitCount = std::numeric_limits<RawType>::digits - 1; // # of exponent bits in a number. static const size_t kExponentBitCount = kBitCount - 1 - kFractionBitCount; // The mask for the sign bit. static const Bits kSignBitMask = static_cast<Bits>(1) << (kBitCount - 1); // The mask for the fraction bits. static const Bits kFractionBitMask = ~static_cast<Bits>(0) >> (kExponentBitCount + 1); // The mask for the exponent bits. static const Bits kExponentBitMask = ~(kSignBitMask | kFractionBitMask); // How many ULP's (Units in the Last Place) we want to tolerate when // comparing two numbers. The larger the value, the more error we // allow. A 0 value means that two numbers must be exactly the same // to be considered equal. // // The maximum error of a single floating-point operation is 0.5 // units in the last place. On Intel CPU's, all floating-point // calculations are done with 80-bit precision, while double has 64 // bits. Therefore, 4 should be enough for ordinary use. // // See the following article for more details on ULP: // http://www.cygnus-software.com/papers/comparingfloats/comparingfloats.htm. static const size_t kMaxUlps = 4; // Constructs a FloatingPoint from a raw floating-point number. // // On an Intel CPU, passing a non-normalized NAN (Not a Number) // around may change its bits, although the new value is guaranteed // to be also a NAN. Therefore, don't expect this constructor to // preserve the bits in x when x is a NAN. explicit FloatingPoint(const RawType& x) { u_.value_ = x; } // Static methods // Reinterprets a bit pattern as a floating-point number. // // This function is needed to test the AlmostEquals() method. static RawType ReinterpretBits(const Bits bits) { FloatingPoint fp(0); fp.u_.bits_ = bits; return fp.u_.value_; } // Returns the floating-point number that represent positive infinity. static RawType Infinity() { return ReinterpretBits(kExponentBitMask); } // Non-static methods // Returns the bits that represents this number. const Bits &bits() const { return u_.bits_; } // Returns the exponent bits of this number. Bits exponent_bits() const { return kExponentBitMask & u_.bits_; } // Returns the fraction bits of this number. Bits fraction_bits() const { return kFractionBitMask & u_.bits_; } // Returns the sign bit of this number. Bits sign_bit() const { return kSignBitMask & u_.bits_; } // Returns true iff this is NAN (not a number). bool is_nan() const { // It's a NAN if the exponent bits are all ones and the fraction // bits are not entirely zeros. return (exponent_bits() == kExponentBitMask) && (fraction_bits() != 0); } // Returns true iff this number is at most kMaxUlps ULP's away from // rhs. In particular, this function: // // - returns false if either number is (or both are) NAN. // - treats really large numbers as almost equal to infinity. // - thinks +0.0 and -0.0 are 0 DLP's apart. bool AlmostEquals(const FloatingPoint& rhs) const { // The IEEE standard says that any comparison operation involving // a NAN must return false. if (is_nan() || rhs.is_nan()) return false; return DistanceBetweenSignAndMagnitudeNumbers(u_.bits_, rhs.u_.bits_) <= kMaxUlps; } private: // The data type used to store the actual floating-point number. union FloatingPointUnion { RawType value_; // The raw floating-point number. Bits bits_; // The bits that represent the number. }; // Converts an integer from the sign-and-magnitude representation to // the biased representation. More precisely, let N be 2 to the // power of (kBitCount - 1), an integer x is represented by the // unsigned number x + N. // // For instance, // // -N + 1 (the most negative number representable using // sign-and-magnitude) is represented by 1; // 0 is represented by N; and // N - 1 (the biggest number representable using // sign-and-magnitude) is represented by 2N - 1. // // Read http://en.wikipedia.org/wiki/Signed_number_representations // for more details on signed number representations. static Bits SignAndMagnitudeToBiased(const Bits &sam) { if (kSignBitMask & sam) { // sam represents a negative number. return ~sam + 1; } else { // sam represents a positive number. return kSignBitMask | sam; } } // Given two numbers in the sign-and-magnitude representation, // returns the distance between them as an unsigned number. static Bits DistanceBetweenSignAndMagnitudeNumbers(const Bits &sam1, const Bits &sam2) { const Bits biased1 = SignAndMagnitudeToBiased(sam1); const Bits biased2 = SignAndMagnitudeToBiased(sam2); return (biased1 >= biased2) ? (biased1 - biased2) : (biased2 - biased1); } FloatingPointUnion u_; }; EDIT: This post is 4 years old. It's probably still valid, and the code is nice, but some people found improvements. Best go get the latest version of AlmostEquals right from the Google Test source code, and not the one I pasted up here.
Comparing floating point numbers for depends on the context. Since even changing the order of operations can produce different results, it is important to know how "equal" you want the numbers to be.
Comparing floating point numbers by Bruce Dawson is a good place to start when looking at floating point comparison.
The following definitions are from The art of computer programming by Knuth:
bool approximatelyEqual(float a, float b, float epsilon) { return fabs(a - b) <= ( (fabs(a) < fabs(b) ? fabs(b) : fabs(a)) * epsilon); } bool essentiallyEqual(float a, float b, float epsilon) { return fabs(a - b) <= ( (fabs(a) > fabs(b) ? fabs(b) : fabs(a)) * epsilon); } bool definitelyGreaterThan(float a, float b, float epsilon) { return (a - b) > ( (fabs(a) < fabs(b) ? fabs(b) : fabs(a)) * epsilon); } bool definitelyLessThan(float a, float b, float epsilon) { return (b - a) > ( (fabs(a) < fabs(b) ? fabs(b) : fabs(a)) * epsilon); } Of course, choosing epsilon depends on the context, and determines how equal you want the numbers to be.
Another method of comparing floating point numbers is to look at the ULP (units in last place) of the numbers. While not dealing specifically with comparisons, the paper What every computer scientist should know about floating point numbers is a good resource for understanding how floating point works and what the pitfalls are, including what ULP is.
For a more in depth approach read Comparing floating point numbers. Here is the code snippet from that link:
// Usable AlmostEqual function bool AlmostEqual2sComplement(float A, float B, int maxUlps) { // Make sure maxUlps is non-negative and small enough that the // default NAN won't compare as equal to anything. assert(maxUlps > 0 && maxUlps < 4 * 1024 * 1024); int aInt = *(int*)&A; // Make aInt lexicographically ordered as a twos-complement int if (aInt < 0) aInt = 0x80000000 - aInt; // Make bInt lexicographically ordered as a twos-complement int int bInt = *(int*)&B; if (bInt < 0) bInt = 0x80000000 - bInt; int intDiff = abs(aInt - bInt); if (intDiff <= maxUlps) return true; return false; } The portable way to get epsilon in C++ is
#include <limits> std::numeric_limits<double>::epsilon() Then the comparison function becomes
#include <cmath> #include <limits> bool AreSame(double a, double b) { return std::fabs(a - b) < std::numeric_limits<double>::epsilon(); } Realizing this is an old thread but this article is one of the most straight forward ones I have found on comparing floating point numbers and if you want to explore more it has more detailed references as well and it the main site covers a complete range of issues dealing with floating point numbers The Floating-Point Guide :Comparison.
We can find a somewhat more practical article in Floating-point tolerances revisited and notes there is absolute tolerance test, which boils down to this in C++:
bool absoluteToleranceCompare(double x, double y) { return std::fabs(x - y) <= std::numeric_limits<double>::epsilon() ; } and relative tolerance test:
bool relativeToleranceCompare(double x, double y) { double maxXY = std::max( std::fabs(x) , std::fabs(y) ) ; return std::fabs(x - y) <= std::numeric_limits<double>::epsilon()*maxXY ; } The article notes that the absolute test fails when x and y are large and fails in the relative case when they are small. Assuming he absolute and relative tolerance is the same a combined test would look like this:
bool combinedToleranceCompare(double x, double y) { double maxXYOne = std::max( { 1.0, std::fabs(x) , std::fabs(y) } ) ; return std::fabs(x - y) <= std::numeric_limits<double>::epsilon()*maxXYOne ; } The code you wrote is bugged :
return (diff < EPSILON) && (-diff > EPSILON); The correct code would be :
return (diff < EPSILON) && (diff > -EPSILON); (...and yes this is different)
I wonder if fabs wouldn't make you lose lazy evaluation in some case. I would say it depends on the compiler. You might want to try both. If they are equivalent in average, take the implementation with fabs.
If you have some info on which of the two float is more likely to be bigger than then other, you can play on the order of the comparison to take better advantage of the lazy evaluation.
Finally you might get better result by inlining this function. Not likely to improve much though...
Edit: OJ, thanks for correcting your code. I erased my comment accordingly
`return fabs(a - b) < EPSILON;
This is fine if:
But otherwise it'll lead you into trouble. Double precision numbers have a resolution of about 16 decimal places. If the two numbers you are comparing are larger in magnitude than EPSILON*1.0E16, then you might as well be saying:
return a==b; I'll examine a different approach that assumes you need to worry about the first issue and assume the second is fine your application. A solution would be something like:
#define VERYSMALL (1.0E-150) #define EPSILON (1.0E-8) bool AreSame(double a, double b) { double absDiff = fabs(a - b); if (absDiff < VERYSMALL) { return true; } double maxAbs = max(fabs(a) - fabs(b)); return (absDiff/maxAbs) < EPSILON; } This is expensive computationally, but it is sometimes what is called for. This is what we have to do at my company because we deal with an engineering library and inputs can vary by a few dozen orders of magnitude.
Anyway, the point is this (and applies to practically every programming problem): Evaluate what your needs are, then come up with a solution to address your needs -- don't assume the easy answer will address your needs. If after your evaluation you find that fabs(a-b) < EPSILON will suffice, perfect -- use it! But be aware of its shortcomings and other possible solutions too.
As others have pointed out, using a fixed-exponent epsilon (such as 0.0000001) will be useless for values away from the epsilon value. For example, if your two values are 10000.000977 and 10000, then there are NO 32-bit floating-point values between these two numbers -- 10000 and 10000.000977 are as close as you can possibly get without being bit-for-bit identical. Here, an epsilon of less than 0.0009 is meaningless; you might as well use the straight equality operator.
Likewise, as the two values approach epsilon in size, the relative error grows to 100%.
Thus, trying to mix a fixed point number such as 0.00001 with floating-point values (where the exponent is arbitrary) is a pointless exercise. This will only ever work if you can be assured that the operand values lie within a narrow domain (that is, close to some specific exponent), and if you properly select an epsilon value for that specific test. If you pull a number out of the air ("Hey! 0.00001 is small, so that must be good!"), you're doomed to numerical errors. I've spent plenty of time debugging bad numerical code where some poor schmuck tosses in random epsilon values to make yet another test case work.
If you do numerical programming of any kind and believe you need to reach for fixed-point epsilons, READ BRUCE'S ARTICLE ON COMPARING FLOATING-POINT NUMBERS.
I ended up spending quite some time going through material in this great thread. I doubt everyone wants to spend so much time so I would highlight the summary of what I learned and the solution I implemented.
Quick Summary
numeric_limits::epsilon() which is same as FLT_EPSILON in float.h. This is however problematic because epsilon for comparing values like 1.0 if not same as epsilon for values like 1E9. The FLT_EPSILON is defined for 1.0.fabs(a-b) <= epsilon however this doesn't work because default epsilon is defined for 1.0. We need to scale epsilon up or down in terms of a and b.max(a,b) or you can get next representable numbers around a and then see if b falls into that range. The former is called "relative" method and later is called ULP method.Utility Functions Implementation (C++11)
//implements relative method - do not use for comparing with zero //use this most of the time, tolerance needs to be meaningful in your context template<typename TReal> static bool isApproximatelyEqual(TReal a, TReal b, TReal tolerance = std::numeric_limits<TReal>::epsilon()) { TReal diff = std::fabs(a - b); if (diff <= tolerance) return true; if (diff < std::fmax(std::fabs(a), std::fabs(b)) * tolerance) return true; return false; } //supply tolerance that is meaningful in your context //for example, default tolerance may not work if you are comparing double with float template<typename TReal> static bool isApproximatelyZero(TReal a, TReal tolerance = std::numeric_limits<TReal>::epsilon()) { if (std::fabs(a) <= tolerance) return true; return false; } //use this when you want to be on safe side //for example, don't start rover unless signal is above 1 template<typename TReal> static bool isDefinitelyLessThan(TReal a, TReal b, TReal tolerance = std::numeric_limits<TReal>::epsilon()) { TReal diff = a - b; if (diff < tolerance) return true; if (diff < std::fmax(std::fabs(a), std::fabs(b)) * tolerance) return true; return false; } template<typename TReal> static bool isDefinitelyGreaterThan(TReal a, TReal b, TReal tolerance = std::numeric_limits<TReal>::epsilon()) { TReal diff = a - b; if (diff > tolerance) return true; if (diff > std::fmax(std::fabs(a), std::fabs(b)) * tolerance) return true; return false; } //implements ULP method //use this when you are only concerned about floating point precision issue //for example, if you want to see if a is 1.0 by checking if its within //10 closest representable floating point numbers around 1.0. template<typename TReal> static bool isWithinPrecisionInterval(TReal a, TReal b, unsigned int interval_size = 1) { TReal min_a = a - (a - std::nextafter(a, std::numeric_limits<TReal>::lowest())) * interval_size; TReal max_a = a + (std::nextafter(a, std::numeric_limits<TReal>::max()) - a) * interval_size; return min_a <= b && max_a >= b; } General-purpose comparison of floating-point numbers is generally meaningless. How to compare really depends on a problem at hand. In many problems, numbers are sufficiently discretized to allow comparing them within a given tolerance. Unfortunately, there are just as many problems, where such trick doesn't really work. For one example, consider working with a Heaviside (step) function of a number in question (digital stock options come to mind) when your observations are very close to the barrier. Performing tolerance-based comparison wouldn't do much good, as it would effectively shift the issue from the original barrier to two new ones. Again, there is no general-purpose solution for such problems and the particular solution might require going as far as changing the numerical method in order to achieve stability.
Unfortunately, even your "wasteful" code is incorrect. EPSILON is the smallest value that could be added to 1.0 and change its value. The value 1.0 is very important ― larger numbers do not change when added to EPSILON. Now, you can scale this value to the numbers you are comparing to tell whether they are different or not. The correct expression for comparing two doubles is:
if (fabs(a - b) <= DBL_EPSILON * fmax(fabs(a), fabs(b))) { // ... } This is at a minimum. In general, though, you would want to account for noise in your calculations and ignore a few of the least significant bits, so a more realistic comparison would look like:
if (fabs(a - b) <= 16 * DBL_EPSILON * fmax(fabs(a), fabs(b))) { // ... } If comparison performance is very important to you and you know the range of your values, then you should use fixed-point numbers instead.
My class based on previously posted answers. Very similar to Google's code but I use a bias which pushes all NaN values above 0xFF000000. That allows a faster check for NaN.
This code is meant to demonstrate the concept, not be a general solution. Google's code already shows how to compute all the platform specific values and I didn't want to duplicate all that. I've done limited testing on this code.
typedef unsigned int U32; // Float Memory Bias (unsigned) // ----- ------ --------------- // NaN 0xFFFFFFFF 0xFF800001 // NaN 0xFF800001 0xFFFFFFFF // -Infinity 0xFF800000 0x00000000 --- // -3.40282e+038 0xFF7FFFFF 0x00000001 | // -1.40130e-045 0x80000001 0x7F7FFFFF | // -0.0 0x80000000 0x7F800000 |--- Valid <= 0xFF000000. // 0.0 0x00000000 0x7F800000 | NaN > 0xFF000000 // 1.40130e-045 0x00000001 0x7F800001 | // 3.40282e+038 0x7F7FFFFF 0xFEFFFFFF | // Infinity 0x7F800000 0xFF000000 --- // NaN 0x7F800001 0xFF000001 // NaN 0x7FFFFFFF 0xFF7FFFFF // // Either value of NaN returns false. // -Infinity and +Infinity are not "close". // -0 and +0 are equal. // class CompareFloat{ public: union{ float m_f32; U32 m_u32; }; static bool CompareFloat::IsClose( float A, float B, U32 unitsDelta = 4 ) { U32 a = CompareFloat::GetBiased( A ); U32 b = CompareFloat::GetBiased( B ); if ( (a > 0xFF000000) || (b > 0xFF000000) ) { return( false ); } return( (static_cast<U32>(abs( a - b ))) < unitsDelta ); } protected: static U32 CompareFloat::GetBiased( float f ) { U32 r = ((CompareFloat*)&f)->m_u32; if ( r & 0x80000000 ) { return( ~r - 0x007FFFFF ); } return( r + 0x7F800000 ); } }; I'd be very wary of any of these answers that involves floating point subtraction (e.g., fabs(a-b) < epsilon). First, the floating point numbers become more sparse at greater magnitudes and at high enough magnitudes where the spacing is greater than epsilon, you might as well just be doing a == b. Second, subtracting two very close floating point numbers (as these will tend to be, given that you're looking for near equality) is exactly how you get catastrophic cancellation.
While not portable, I think grom's answer does the best job of avoiding these issues.
There are actually cases in numerical software where you want to check whether two floating point numbers are exactly equal. I posted this on a similar question
https://stackoverflow.com/a/10973098/1447411
So you can not say that "CompareDoubles1" is wrong in general.
It depends on how precise you want the comparison to be. If you want to compare for exactly the same number, then just go with ==. (You almost never want to do this unless you actually want exactly the same number.) On any decent platform you can also do the following:
diff= a - b; return fabs(diff)<EPSILON; as fabs tends to be pretty fast. By pretty fast I mean it is basically a bitwise AND, so it better be fast.
And integer tricks for comparing doubles and floats are nice but tend to make it more difficult for the various CPU pipelines to handle effectively. And it's definitely not faster on certain in-order architectures these days due to using the stack as a temporary storage area for values that are being used frequently. (Load-hit-store for those who care.)
In terms of the scale of quantities:
If epsilon is the small fraction of the magnitude of quantity (i.e. relative value) in some certain physical sense and A and B types is comparable in the same sense, than I think, that the following is quite correct:
#include <limits> #include <iomanip> #include <iostream> #include <cmath> #include <cstdlib> #include <cassert> template< typename A, typename B > inline bool close_enough(A const & a, B const & b, typename std::common_type< A, B >::type const & epsilon) { using std::isless; assert(isless(0, epsilon)); // epsilon is a part of the whole quantity assert(isless(epsilon, 1)); using std::abs; auto const delta = abs(a - b); auto const x = abs(a); auto const y = abs(b); // comparable generally and |a - b| < eps * (|a| + |b|) / 2 return isless(epsilon * y, x) && isless(epsilon * x, y) && isless((delta + delta) / (x + y), epsilon); } int main() { std::cout << std::boolalpha << close_enough(0.9, 1.0, 0.1) << std::endl; std::cout << std::boolalpha << close_enough(1.0, 1.1, 0.1) << std::endl; std::cout << std::boolalpha << close_enough(1.1, 1.2, 0.01) << std::endl; std::cout << std::boolalpha << close_enough(1.0001, 1.0002, 0.01) << std::endl; std::cout << std::boolalpha << close_enough(1.0, 0.01, 0.1) << std::endl; return EXIT_SUCCESS; } /// testing whether two doubles are almost equal. We consider two doubles /// equal if the difference is within the range [0, epsilon). /// /// epsilon: a positive number (supposed to be small) /// /// if either x or y is 0, then we are comparing the absolute difference to /// epsilon. /// if both x and y are non-zero, then we are comparing the relative difference /// to epsilon. bool almost_equal(double x, double y, double epsilon) { double diff = x - y; if (x != 0 && y != 0){ diff = diff/y; } if (diff < epsilon && -1.0*diff < epsilon){ return true; } return false; } I used this function for my small project and it works, but note the following:
Double precision error can create a surprise for you. Let's say epsilon = 1.0e-6, then 1.0 and 1.000001 should NOT be considered equal according to the above code, but on my machine the function considers them to be equal, this is because 1.000001 can not be precisely translated to a binary format, it is probably 1.0000009xxx. I test it with 1.0 and 1.0000011 and this time I get the expected result.
I use this code:
bool AlmostEqual(double v1, double v2) { return (std::fabs(v1 - v2) < std::fabs(std::min(v1, v2)) * std::numeric_limits<double>::epsilon()); } I write this for java, but maybe you find it useful. It uses longs instead of doubles, but takes care of NaNs, subnormals, etc.
public static boolean equal(double a, double b) { final long fm = 0xFFFFFFFFFFFFFL; // fraction mask final long sm = 0x8000000000000000L; // sign mask final long cm = 0x8000000000000L; // most significant decimal bit mask long c = Double.doubleToLongBits(a), d = Double.doubleToLongBits(b); int ea = (int) (c >> 52 & 2047), eb = (int) (d >> 52 & 2047); if (ea == 2047 && (c & fm) != 0 || eb == 2047 && (d & fm) != 0) return false; // NaN if (c == d) return true; // identical - fast check if (ea == 0 && eb == 0) return true; // ±0 or subnormals if ((c & sm) != (d & sm)) return false; // different signs if (abs(ea - eb) > 1) return false; // b > 2*a or a > 2*b d <<= 12; c <<= 12; if (ea < eb) c = c >> 1 | sm; else if (ea > eb) d = d >> 1 | sm; c -= d; return c < 65536 && c > -65536; // don't use abs(), because: // There is a posibility c=0x8000000000000000 which cannot be converted to positive } public static boolean zero(double a) { return (Double.doubleToLongBits(a) >> 52 & 2047) < 3; } Keep in mind that after a number of floating-point operations, number can be very different from what we expect. There is no code to fix that.
My way may not be correct but useful
Convert both float to strings and then do string compare
bool IsFlaotEqual(float a, float b, int decimal) { TCHAR form[50] = _T(""); _stprintf(form, _T("%%.%df"), decimal); TCHAR a1[30] = _T(""), a2[30] = _T(""); _stprintf(a1, form, a); _stprintf(a2, form, b); if( _tcscmp(a1, a2) == 0 ) return true; return false; } operator overlaoding can also be done
You cannot compare two double with a fixed EPSILON. Depending on the value of double, EPSILON varies.
A better double comparison would be:
bool same(double a, double b) { return std::nextafter(a, std::numeric_limits<double>::lowest()) <= b && std::nextafter(a, std::numeric_limits<double>::max()) >= b; } In a more generic way:
template <typename T> bool compareNumber(const T& a, const T& b) { return std::abs(a - b) < std::numeric_limits<T>::epsilon(); } Why not perform bitwise XOR? Two floating point numbers are equal if their corresponding bits are equal. I think, the decision to place the exponent bits before mantissa was made to speed up comparison of two floats. I think, many answers here are missing the point of epsilon comparison. Epsilon value only depends on to what precision floating point numbers are compared. For example, after doing some arithmetic with floats you get two numbers: 2.5642943554342 and 2.5642943554345. They are not equal, but for the solution only 3 decimal digits matter so then they are equal: 2.564 and 2.564. In this case you choose epsilon equal to 0.001. Epsilon comparison is also possible with bitwise XOR. Correct me if I am wrong.