IEEE floating-point exceptions in C++

This page will answer the following questions.

My program just printed out 1.#IND or 1.#INF (on Windows) or nan or inf (on Linux). What happened?
How can I tell if a number is really a number and not a NaN or an infinity?
How can I find out more details at runtime about kinds of NaNs and infinities?
Do you have any sample code to show how this works?
Where can I learn more?

These questions have to do with floating point exceptions. If you get some strange non-numeric output where you're expecting a number, you've either exceeded the finite limits of floating point arithmetic or you've asked for some result that is undefined. To keep things simple, I'll stick to working with the double floating point type. Similar remarks hold for float types.

Debugging 1.#IND, 1.#INF, nan, and inf

If your operation would generate a larger positive number than could be stored in a double, the operation will return 1.#INF on Windows or inf on Linux. Similarly your code will return 11.#INF or -inf if the result would be a negative number too large to store in a double. Dividing a positive number by zero produces a positive infinity and dividing a negative number by zero produces a negative infinity. Example code at the end of this page will demonstrate some operations that produce infinities.

Some operations don't make mathematical sense, such as taking the square root of a negative number. (Yes, this operation makes sense in the context of complex numbers, but a double represents a real number and so there is no double to represent the result.) The same is true for logarithms of negative numbers. Both sqrt(-1.0) and log(-1.0) would return a NaN, the generic term for a "number" that is "not a number". Windows displays a NaN as -1.#IND ("IND" for "indeterminate") while Linux displays nan. Other operations that would return a NaN include 0/0, 0*∞, and ∞/∞. See the sample code below for examples.

In short, if you get 1.#INF or inf, look for overflow or division by zero. If you get 1.#IND or nan, look for illegal operations. Maybe you simply have a bug. If it's more subtle and you have something that is difficult to compute, see Avoiding Overflow, Underflow, and Loss of Precision. That article gives tricks for computing results that have intermediate steps overflow if computed directly.

Testing for NaNs and infinities

Next suppose you want to test whether a number is an infinity or a NaN. For example, you may want to write to a log file print a debug message when a numerical result goes bad, or you may want to execute some sort of alternate logic in your code. There are simple, portable ways to get summary information and more complicated, less portable ways to get more information.

First, the simple solution. If you want to test whether a double variable contains a valid number, you can check whether x == x. This looks like it should always be true, but it's not! Ordinary numbers always equal themselves, but NaNs do not. I've used this trick on Windows, Linux, and Mac OSX. If you ever use this trick, put big bold comments around your code so that some well-meaning person won't come behind you and delete what he or she things is useless code. Better yet, put the test in a well-documented function in a library that has controlled access. The following function will test whether x is a (possibly infinite) number.

             bool IsNumber(double x)      {         // This looks like it should always be true,          // but it's false if x is a NaN.         return (x == x);      }

To test whether a variable contains a finite number, (i.e. not a NaN and not an infinity) you can use code like the following.

             bool IsFiniteNumber(double x)      {         return (x <= DBL_MAX && x >= -DBL_MAX);      }

Here DBL_MAX is a constant defined in float.h as the largest double that can be represented. Comparisons with NaNs always fail, even when comparing to themselves, and so the test above will fail for a NaN. If x is not a NaN but is infinite, one of the two tests will fail depending on whether it is a positive infinity or negative infinity.

Getting more information programmatically

To get more detail about the type of a floating point number, there is a function _fpclass on Windows and a corresponding function fp_class_d on Linux. I have not been able to get the corresponding Linux code to work and so I'll stick to what I've tested and just talk about Windows from here on out.

The Windows function _fpclass returns one of the following values:

        _FPCLASS_SNAN   // signaling NaN         _FPCLASS_QNAN   // quiet NaN         _FPCLASS_NINF   // negative infinity         _FPCLASS_NN     // negative normal         _FPCLASS_ND     // negative denormal         _FPCLASS_NZ     // -0         _FPCLASS_PZ     // +0         _FPCLASS_PD     // positive denormal         _FPCLASS_PN     // positive normal         _FPCLASS_PINF   // positive infinity

The following code illustrates which kinds of operations result in which kinds of numbers. To port this code to Linux, the FPClass function would need to use fp_class_d and its corresponding constants.

        #include          #include          #include          #include           using namespace std;          string FPClass(double x)         {             int i = _fpclass(x);             string s;             switch (i)             {             case _FPCLASS_SNAN: s = "Signaling NaN";                break;             case _FPCLASS_QNAN: s = "Quiet NaN";                    break;              case _FPCLASS_NINF: s = "Negative infinity (-INF)";     break;              case _FPCLASS_NN:   s = "Negative normalized non-zero"; break;             case _FPCLASS_ND:   s = "Negative denormalized";        break;              case _FPCLASS_NZ:   s = "Negative zero (-0)";           break;              case _FPCLASS_PZ:   s = "Positive 0 (+0)";              break;              case _FPCLASS_PD:   s = "Positive denormalized";        break;              case _FPCLASS_PN:   s = "Positive normalized non-zero"; break;              case _FPCLASS_PINF: s = "Positive infinity (+INF)";     break;             }             return s;         }          string HexDump(double x)         {             unsigned long* pu;             pu = (unsigned long*)&x;             ostringstream os;             os << hex << pu[0] << " " << pu[1];             return os.str();         }          // ----------------------------------------------------------------------------         int main()         {             double x, y, z;              cout << "Testing z = 1/0\n";             // cannot set x = 1/0 directly or would produce compile error.             x = 1.0; y = 0; z = x/y;             cout << "z = " << x/y << "\n";             cout << HexDump(z) << " _fpclass(z) = " << FPClass(z) << "\n";              cout << "\nTesting z = -1/0\n";             x = -1.0; y = 0; z = x/y;             cout << "z = " << x/y << "\n";             cout << HexDump(z) << " _fpclass(z) = " << FPClass(z) << "\n";              cout << "\nTesting z = sqrt(-1)\n";             x = -1.0;             z = sqrt(x);             cout << "z = " << z << "\n";             cout << HexDump(z) << " _fpclass(z) = " << FPClass(z) << "\n";              cout << "\nTesting z = log(-1)\n";             x = -1.0;             z = sqrt(x);             cout << "z = " << z << "\n";             cout << HexDump(z) << " _fpclass(z) = " << FPClass(z) << "\n";              cout << "\nTesting overflow\n";             z = DBL_MAX;             cout << "z = DBL_MAX = " << z;              z *= 2.0;             cout << "; 2z = " << z << "\n";             cout << HexDump(z) << " _fpclass(z) = " << FPClass(z) << "\n";              cout << "\nTesting denormalized underflow\n";             z = DBL_MIN;             cout << "z = DBL_MIN = " << z << "\n";             cout << HexDump(z) << " _fpclass(z) = " << FPClass(z) << "\n";             z /= pow(2.0, 52);             cout << "z = DBL_MIN / 2^52= " << z << "\n";             cout << HexDump(z) << " _fpclass(z) = " << FPClass(z) << "\n";             z /= 2;             cout << "z = DBL_MIN / 2^53= " << z << "\n";             cout << HexDump(z) << " _fpclass(z) = " << FPClass(z) << "\n";              cout << "\nTesting z = +infinity + -infinty\n";             x = 1.0; y = 0.0; x /= y; y = -x;             cout << x << " + " << y << " = " << z << "\n";             cout << HexDump(z) << " _fpclass(z) = " << FPClass(z) << "\n";              cout << "\nTesting z = 0 * infinity\n";             x = 1.0; y = 0.0; x /= y; z = 0.0*x;             cout << "x = " << x << "; z = 0*x = " << z << "\n";             cout << HexDump(z) << " _fpclass(z) = " << FPClass(z) << "\n";              cout << "\nTesting 0/0\n";             x = 0.0; y = 0.0; z = x/y;             cout << "z = 0/0 = " << z << "\n";             cout << HexDump(z) << " _fpclass(z) = " << FPClass(z) << "\n";              cout << "\nTesting z = infinity/infinity\n";             x = 1.0; y = 0.0; x /= y; y = x; z = x/y;             cout << "x = " << x << "; z = x/x = " << z << "\n";             cout << HexDump(z) << " _fpclass(z) = " << FPClass(z) << "\n";              cout << "\nTesting x fmod 0\n";             x = 1.0; y = 0.0; z = fmod(x, y);             cout << "fmod(" << x << ", " << y << ") = " << z << "\n";             cout << HexDump(z) << " _fpclass(z) = " << FPClass(z) << "\n";              cout << "\nTesting infinity fmod x\n";             y = 1.0; x = 0.0; y /= x; z = fmod(y, x);             cout << "fmod(" << y << ", " << x << ") = " << z << "\n";             cout << HexDump(z) << " _fpclass(z) = " << FPClass(z) << "\n";              cout << "\nGetting cout to print QNAN\n";             unsigned long nan[2]={0xffffffff, 0x7fffffff};             z = *( double* )nan;             cout << "z = " << z << "\n";             cout << HexDump(z) << " _fpclass(z) = " << FPClass(z) << "\n";              return 0;         }

To learn more

For much more detail regarding exceptions and IEEE arithmetic in general, see What every computer scientist should know about floating-point arithmetic.

Other C++ articles: random number generation, strings, regular expressions

1. 요약
Anti Cracking 관련된 자료를 제공하는 사이트

2. 본문
1) CrackZ's Reverse Engineering Page (1997-2007)
Link : http://www.woodmann.com/crackz/Miscpapers.htm

2) Reverse Engineering Team
Link : http://www.reteam.org/index.html

3) rEVERSE
Link : http://ampm.ddns.co.kr/~reverse/

4) 여리의 작업실
Link : http://www.zap.pe.kr/

5) The Reverse Code Engineering Community
Link : http://www.reverse-engineering.net/

6) Secure Software Engineering
Link : http://www.secure-software-engineering.com/

7) Security Labs
Link : http://securitylabs.websense.com/content/Blogs/2491.aspx

8) OpenRCE Anti Reverse Engineering Techniques Database
Desc : 어셈블리 코드를 직접 예제로 보여주면서 설명해 주는 사이트, 실질적인 코드를 제공
Link : http://www.openrce.org/reference_library/anti_reversing

9) Scan of the Month 33: Anti Reverse Engineering Uncovered
Desc : Anti Reverse Engineering 기법을 한 페이지에 잘 정리하였음
Link : http://www.honeynet.org/scans/scan33/nico/index.html

10) A Survey of Reverse Engineering Tools
Link : http://www.cs.drexel.edu/~spiros/teaching/CS675/asmrceFINAL.pdf

11) Honeynet Project
Link : http://www.honeynet.org/

12) Process Stalker
Desc : 프로세스의 실행 경로를 탐색하며 문제를 파악하는 방법
Link : http://www.openrce.org/articles/full_view/12

13) OllyDbg
Link : http://ollydbg.de/

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