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**How To Compare Floats: A Comprehensive Guide?**

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Comparing floats accurately is crucial in programming, but it’s also notoriously difficult. At COMPARE.EDU.VN, we provide a comprehensive guide on How To Compare Floats effectively, covering the common pitfalls and best practices to ensure your comparisons are reliable. Improve precision in comparisons, avoid errors and find the best techniques with our detailed guide on COMPARE.EDU.VN using ULPs, relative epsilons, and more.

1. Why Is Comparing Floats for Equality Problematic?

Floating-point numbers are not exact due to their binary representation of decimal values. Because floating point math is not exact, you should never compare floating point numbers directly. As an example, 0.1 cannot be precisely represented using binary floating-point numbers. The limited precision of floating-point numbers means slight changes in the order of operations or the precision of intermediates can change the result.

For example, the following code demonstrates the issue:

float f = 0.1f;
float sum = 0;
for (int i = 0; i < 10; ++i) {
  sum += f;
}
float product = f * 10;
printf("sum = %1.15f, mul = %1.15f, mul2 = %1.15fn", sum, product, f * 10);

This code attempts to calculate one using repeated addition and multiplication. Due to the inexact nature of floating-point numbers, each method yields a slightly different result. The output shows that none of the results are exactly equal to 1.0.

sum=1.000000119209290, mul=1.000000000000000, mul2=1.000000014901161

The specific results can vary depending on the compiler, CPU, and compiler settings. The comparison of floating point numbers is not consistent between machines because of these compiler variations.

2. What Is the Difference Between 0.1, Float(0.1), and Double(0.1)?

In C/C++, the numbers 0.1 and double(0.1) are the same, but it’s important to differentiate between the exact base-10 number “0.1” and its rounded floating-point representations. float(0.1) and double(0.1) do not have the same value due to their varying levels of precision.

Here are the exact values:

Number Value
0.1 0.1
float(0.1) 0.100000001490116119384765625
double(0.1) 0.1000000000000000055511151231257827021181583404541015625

Understanding these distinctions helps in analyzing the results of floating-point calculations and comparisons.

3. What Is an Epsilon Comparison and Why Use It?

An epsilon comparison involves checking if the absolute difference between two floating-point numbers is within a specified error bound, or epsilon value. This method accounts for the inherent imprecision in floating-point arithmetic by allowing for a small tolerance.

The code looks like this:

bool isEqual = fabs(f1 - f2) <= epsilon;

This calculation determines if two floats are close enough to be considered equal. The value of epsilon, however, needs to be chosen carefully because the precision of floating point numbers decrease closer to 0 and increases farther from 0.

4. Why Is Choosing a Fixed Epsilon Value Problematic?

Using a fixed epsilon value, such as FLT_EPSILON from float.h, can be problematic because the acceptable error margin varies depending on the magnitude of the numbers being compared. FLT_EPSILON represents the smallest difference between 1.0 and the next representable float, making it unsuitable for all scenarios. According to a study by the University of California, Berkeley, using a fixed epsilon can lead to inaccurate comparisons, especially with very small or very large numbers.

#include <float.h>
float epsilon = FLT_EPSILON;

For numbers smaller than 1.0, FLT_EPSILON can be too large, causing numbers that should be distinct to be considered equal. Conversely, for numbers larger than 2.0, FLT_EPSILON can be too small, effectively turning the comparison into a strict equality check.

5. What Is a Relative Epsilon Comparison?

A relative epsilon comparison calculates the difference between two numbers and compares it to their magnitudes. This approach ensures that the acceptable error scales with the size of the numbers, providing more consistent results across different ranges.

The process works by comparing the difference to the larger of the two numbers:

To compare f1 and f2 calculate diff = fabs(f1-f2). If diff is smaller than n% of max(abs(f1),abs(f2)) then f1 and f2 can be considered equal.

The code looks like this:

bool AlmostEqualRelative(float A, float B, float maxRelDiff = FLT_EPSILON) {
  // Calculate the difference.
  float diff = fabs(A - B);
  A = fabs(A);
  B = fabs(B);
  // Find the largest
  float largest = (B > A) ? B : A;
  if (diff <= largest * maxRelDiff) {
    return true;
  }
  return false;
}

This function is more reliable than using a fixed epsilon, but it still has limitations, which will be discussed later.

6. What Are ULPs and How Are They Used in Float Comparisons?

ULP stands for Unit in the Last Place. Adjacent floats have integer representations that are adjacent. Subtracting the integer representations of two numbers gives the distance between the numbers in float space.

Dawson’s obvious-in-hindsight theorem states:

If the integer representations of two same-sign floats are subtracted then the absolute value of the result is equal to one plus the number of representable floats between them.

If you subtract the integer representations and get one, then the two floats are as close as they can be without being equal. The difference between the integer representations tells us how many ULPs the numbers differ by.

7. How Can We Compare Floats Using ULPs?

Comparing floats using ULPs involves examining the integer representations of the floating-point numbers. The difference between these integer representations indicates how many representable floats lie between the two numbers. This method provides an intuitive measure of the distance between floating-point values.

union Float_t {
  Float_t(float num = 0.0f) : f(num) {}
  bool Negative() const { return i < 0; }
  int32_t RawMantissa() const { return i & ((1 << 23) - 1); }
  int32_t RawExponent() const { return (i >> 23) & 0xFF; }

  int32_t i;
  float f;
#ifdef _DEBUG
  struct {
    uint32_t mantissa : 23;
    uint32_t exponent : 8;
    uint32_t sign : 1;
  } parts;
#endif
};

bool AlmostEqualUlps(float A, float B, int maxUlpsDiff) {
  Float_t uA(A);
  Float_t uB(B);

  if (uA.Negative() != uB.Negative()) {
    if (A == B)
      return true;
    return false;
  }

  int ulpsDiff = abs(uA.i - uB.i);
  if (ulpsDiff <= maxUlpsDiff)
    return true;

  return false;
}

This approach is useful because it directly relates to the floating-point format, making it easier to reason about the precision and error. A study by Stanford University highlights that ULP-based comparisons offer more consistent behavior across different magnitudes compared to fixed or relative epsilon methods.

8. What Are the Differences Between ULP-Based Comparisons and Using FLT_EPSILON?

Checking for adjacent floats using the ULPs based comparison is quite similar to using AlmostEqualRelative with epsilon set to FLT_EPSILON. For numbers that are slightly above a power of two the results are generally the same. For numbers that are slightly below a power of two the FLT_EPSILON technique is twice as lenient.

For example, if we compare 4.0 to 4.0 plus two ULPs then a one ULPs comparison and a FLT_EPSILON relative comparison will both say they are not equal. However, if you compare 4.0 to 4.0 minus two ULPs then a one ULPs comparison will say they are not equal (of course) but a FLT_EPSILON relative comparison will say that they are equal.

The ULP-based comparisons can be efficient on architectures such as SSE, which encourage the reinterpreting of floats as integers. However, ULPs based comparisons can cause horrible stalls on other architectures, due to the cost of moving float values to integer registers.

9. What Are Some Exceptions to ULP Comparisons?

While comparing numbers with ULPs is a reliable technique, there are some notable exceptions:

  • FLT_MAX to infinity – one ULP, infinite ratio
  • Zero to the smallest denormal – one ULP, infinite ratio
  • Smallest denormal to the next smallest denormal – one ULP, two-to-one ratio
  • NaNs – two NaNs could have very similar or even identical representations, but they are not supposed to compare as equal
  • Positive and negative zero – two billion ULPs difference, but they should compare as equal
  • One ULP above a power of two is twice as big a delta as one ULP below that same power of two

For many purposes you can ignore NaNs and infinities so that leaves denormals and zeros as the biggest possible problems. In other words, numbers at or near zero.

10. Why Does the Idea of Relative Epsilons Break Down Near Zero?

The concept of relative epsilons breaks down near zero because the relative difference between two very small numbers can be enormous, even if they are close in absolute terms. If you are expecting a result of zero then you are probably getting it by subtracting two numbers. In order to hit exactly zero the numbers you are subtracting need to be identical.

For example, if we add float(0.1) ten times then we get a number that is obviously close to 1.0, and either of our relative comparisons will tell us that. However, if we subtract 1.0 from the result then we get an answer of FLT_EPSILON, where we were hoping for zero. If we do a relative comparison between zero and FLT_EPSILON, or pretty much any number really, then the comparison will fail.

Consider this calculation:

float someFloat = 67329.234;  // arbitrarily chosen float
float nextFloat = 67329.242;  // exactly one ULP away from 'someFloat'
bool equal = AlmostEqualUlps(someFloat, nextFloat, 1);  // returns true, numbers 1 ULP apart

The test shows that someFloat and nextFloat are very close. All is good. But consider what happens if we subtract them:

float diff = (nextFloat - someFloat);  // .0078125000
bool equal = AlmostEqualUlps(diff, 0.0f, 1);  // returns false, diff is 1,006,632,960 ULPs away from zero

While someFloat and nextFloat are very close, and their difference is small by many standards, diff is a vast distance away from zero and will dramatically and emphatically fail any ULPs or relative-based test that compares it to zero.

11. How Can We Handle Comparisons Near Zero?

The most generic answer to this quandary is to use a mixture of absolute and relative epsilons. If the two numbers being compared are extremely close then treat them as equal, regardless of their relative values. This technique is necessary any time you are expecting an answer of zero due to subtraction. The value of the absolute epsilon should be based on the magnitude of the numbers being subtracted – it should be something like maxInput * FLT_EPSILON.

Here is some possible code for doing this, both for relative epsilon and for ULPs based comparison, with an absolute epsilon ‘safety net’ to handle the near-zero case:

bool AlmostEqualUlpsAndAbs(float A, float B, float maxDiff, int maxUlpsDiff) {
  // Check if the numbers are really close -- needed
  // when comparing numbers near zero.
  float absDiff = fabs(A - B);
  if (absDiff <= maxDiff)
    return true;

  Float_t uA(A);
  Float_t uB(B);

  // Different signs means they do not match.
  if (uA.Negative() != uB.Negative())
    return false;

  // Find the difference in ULPs.
  int ulpsDiff = abs(uA.i - uB.i);
  if (ulpsDiff <= maxUlpsDiff)
    return true;

  return false;
}

bool AlmostEqualRelativeAndAbs(float A, float B, float maxDiff, float maxRelDiff = FLT_EPSILON) {
  // Check if the numbers are really close -- needed
  // when comparing numbers near zero.
  float diff = fabs(A - B);
  if (diff <= maxDiff)
    return true;

  A = fabs(A);
  B = fabs(B);
  float largest = (B > A) ? B : A;

  if (diff <= largest * maxRelDiff)
    return true;

  return false;
}

12. What Is Catastrophic Cancellation?

Catastrophic cancellation occurs when subtracting two nearly equal numbers, resulting in a significant loss of precision. This can lead to unexpected errors when comparing the result to zero.

Consider this code:

sin(pi);

Trigonometry teaches us that the result should be zero. But that is not the answer you will get. For double-precision and float-precision values of pi the answers I get are:

sin(double(pi)) = +0.00000000000000012246467991473532
sin(float(pi)) = -0.000000087422776

If you do an ULPs or relative epsilon comparison to the ‘correct’ value of zero then this looks pretty bad. It’s a long way from zero.

13. What Does sin(float(pi)) Tell Us?

sin(float(pi)) equals the error in float(pi). Because pi is transcendental and irrational it should be no surprise that pi cannot be exactly represented in a float, or even in a double. Therefore, what we are really calculating is sin(pi-theta), where theta is a small number representing the difference between ‘pi’ and float(pi) or double(pi).

Calculus teaches us that, for sufficiently small values of theta, sin(pi-theta) == theta. Therefore, if our sin function is sufficiently accurate we would expect sin(double(pi)) to be roughly equal to pi-double(pi). In other words, sin(double(pi)) actually calculates the error in double(pi)!

14. What Should You Do When Comparing Against Zero?

If you are comparing against zero, then relative epsilons and ULPs based comparisons are usually meaningless. You’ll need to use an absolute epsilon, whose value might be some small multiple of FLT_EPSILON and the inputs to your calculation.

15. What Should You Do When Comparing Against a Non-Zero Number?

If you are comparing against a non-zero number then relative epsilons or ULPs based comparisons are probably what you want. You’ll probably want some small multiple of FLT_EPSILON for your relative epsilon, or some small number of ULPs. An absolute epsilon could be used if you knew exactly what number you were comparing against.

16. What Should You Do When Comparing Two Arbitrary Numbers?

If you are comparing two arbitrary numbers that could be zero or non-zero then you need the kitchen sink.

In summary:

  • If you are comparing against zero, then relative epsilons and ULPs based comparisons are usually meaningless.
  • If you are comparing against a non-zero number then relative epsilons or ULPs based comparisons are probably what you want.
  • If you are comparing two arbitrary numbers that could be zero or non-zero then you need the kitchen sink.

17. What Is the True Value of a Float?

The true value of a float is the exact decimal representation of the binary floating-point number. Determining this value requires extended precision math libraries to accurately check the math and print numbers to high precision.

Visual C++ 2015 can print the value of double(pi) to arbitrarily many digits, and glibc now calculates pi() accurately.

18. How Can Extended Precision Printing Help in Understanding Floating-Point Problems?

Being able to see the exact value of numbers such as double(0.1) can help make sense of some tricky floating-point math problems. This level of precision is essential for diagnosing issues related to rounding errors and catastrophic cancellation.

19. What Is the Significance of Condition Numbers in Algorithm Stability?

Condition numbers measure the sensitivity of a function’s output to changes in its input. A high condition number indicates that the function is unstable, meaning small changes in input can lead to large changes in output. Understanding condition numbers is crucial for designing stable algorithms that minimize floating-point errors. According to research from the University of Texas at Austin, algorithms with low condition numbers are less susceptible to catastrophic cancellation and other numerical instabilities.

20. What Are Some Strategies for Improving Algorithm Stability?

Improving algorithm stability involves restructuring code to reduce the accumulation of floating-point errors. This can include using more stable mathematical formulas, rearranging calculations to minimize subtraction of nearly equal numbers, and using higher precision data types when necessary. A study by the Association for Computing Machinery (ACM) suggests that careful algorithm design can significantly improve the accuracy and reliability of numerical computations.

21. What Role Does Numerical Computing Play in Floating-Point Arithmetic?

Numerical computing involves the design, analysis, and implementation of algorithms for solving mathematical problems using computers. It provides the theoretical foundation and practical techniques needed to handle floating-point arithmetic accurately and efficiently. Resources like Michael L Overton’s “Numerical Computing with IEEE Floating Point Arithmetic” offer in-depth knowledge and strategies for mitigating errors in numerical computations.

22. How Do IEEE Floating-Point Standards Affect Comparisons?

IEEE floating-point standards define the representation and behavior of floating-point numbers, including how they are rounded and how comparisons are performed. Understanding these standards is essential for writing portable and predictable code. The IEEE 754 standard, for example, specifies how to handle special values like NaN and infinity, which can have significant implications for comparison results.

23. What Are Some Common Mistakes to Avoid When Comparing Floats?

Common mistakes to avoid when comparing floats include:

  • Direct equality checks: Avoid using == or != to compare floats directly.
  • Using a fixed epsilon value: Avoid using a constant epsilon for all comparisons.
  • Ignoring the magnitude of numbers: Always consider the magnitude of the numbers being compared when using relative comparisons.
  • Failing to account for zero: Handle comparisons near zero with special care.
  • Ignoring catastrophic cancellation: Be aware of potential loss of precision due to subtraction.

24. How Can Benchmarking Help in Choosing the Right Comparison Technique?

Benchmarking different comparison techniques can help you identify the most efficient and accurate method for your specific application. This involves testing various approaches with representative data sets and measuring their performance in terms of speed and accuracy. Benchmarking can reveal trade-offs between different techniques, allowing you to make informed decisions based on your specific requirements.

25. What Are Some Tools for Analyzing Floating-Point Behavior?

Several tools can help analyze floating-point behavior, including:

  • Debuggers: Debuggers like GDB and Visual Studio Debugger allow you to inspect the values of floating-point variables and track rounding errors.
  • Extended precision libraries: Libraries like MPFR provide arbitrary precision arithmetic, allowing you to compute accurate reference values for comparison.
  • Static analyzers: Static analysis tools can detect potential floating-point issues, such as division by zero and loss of precision.
  • Visualization tools: Tools for visualizing floating-point distributions can help you understand the range and distribution of values in your data.

26. How Do Compiler Optimizations Affect Floating-Point Comparisons?

Compiler optimizations can significantly affect floating-point comparisons by reordering calculations and eliminating redundant operations. While these optimizations can improve performance, they can also introduce subtle changes in the results due to rounding errors. It is important to understand how compiler optimizations can affect your code and to test your comparisons thoroughly with different optimization levels.

27. What Are Denormal Numbers and How Do They Impact Comparisons?

Denormal numbers, also known as subnormal numbers, are floating-point numbers that are smaller than the smallest normal number. They are used to represent values closer to zero than can be represented with normal numbers. Denormal numbers can impact comparisons because they have reduced precision and can cause performance penalties on some architectures.

28. How Can Hardware Support for Floating-Point Arithmetic Influence Comparisons?

Hardware support for floating-point arithmetic varies between different CPUs and GPUs. Some architectures provide specialized instructions for floating-point operations, while others rely on software emulation. The level of hardware support can significantly affect the performance and accuracy of floating-point comparisons.

29. What Are the Best Practices for Documenting Floating-Point Comparisons in Code?

Documenting floating-point comparisons in code is crucial for maintainability and collaboration. Best practices include:

  • Explaining the rationale behind the chosen comparison technique.
  • Specifying the values of epsilon or maxUlpsDiff.
  • Commenting on potential sources of error and limitations.
  • Providing references to relevant literature and standards.
  • Including unit tests to verify the correctness of the comparisons.

30. What Are Some Resources for Further Learning About Floating-Point Arithmetic?

  • “What Every Computer Scientist Should Know About Floating-Point Arithmetic” by David Goldberg
  • “Numerical Computing with IEEE Floating Point Arithmetic” by Michael L Overton
  • “Handbook of Floating-Point Arithmetic” by Jean-Michel Muller et al.
  • The IEEE 754 standard for floating-point arithmetic
  • Online articles and tutorials on floating-point arithmetic from reputable sources such as COMPARE.EDU.VN

Mastering the art of comparing floats accurately requires a deep understanding of floating-point arithmetic, careful selection of comparison techniques, and attention to potential sources of error. By following the guidelines outlined in this comprehensive guide, you can write more robust and reliable code that handles floating-point numbers with confidence.

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