Introduction

I'm writing this blog because of the large number of blogs asking about why they get strange floating arithmetic behaviour in C++. For example:

"WA using GNU C++17 (64) and AC using GNU C++17" https://codeforces.me/blog/entry/78094

"The curious case of the pow function" https://codeforces.me/blog/entry/21844

"Why does this happen?" https://codeforces.me/blog/entry/51884

"Why can this code work strangely?" https://codeforces.me/blog/entry/18005

and many many more.

Example

Here is a simple example of the kind of weird behaviour I'm talking about

#include <iostream>
using namespace std;
 
double f(double a, double b) {
    return a * a - b;
}

int main() {
  cout.precision(60);
  
  // Calculate 10*10 - 1e-15
  double ans;
  ans = f(atof("10"), atof("1e-15"));
  cout << (double)      ans << '\n';
  cout << (int)         ans << "\n\n";
  
  ans = f(atof("10"), atof("1e-15"));
  cout << (int)         ans << '\n';
  cout << (double)      ans << "\n\n";
  
  ans = f(atof("10"), atof("1e-15"));
  cout << (double)      ans << '\n';
  cout << (long double) ans << "\n\n";
  
  ans = f(atof("10"), atof("1e-15"));
  cout << (long double) ans << '\n';
  cout << (double)      ans << "\n\n";
  return 0;
}

100
100

99
100

100
100

99.99999999999999900079927783735911361873149871826171875
100

100
100

100
100

100
100

100
100

Looking at this example, the output that one would expect from $$$10 * 10 - 10^{-15}$$$ is exactly $$$100$$$ since $$$100$$$ is the closest representable value of a double. This is exactly what happens in 64 bit g++. However, in 32 bit g++ there seems to be some kind of hidden excess precision causing the output to only sometimes(???) be $$$100$$$.

Explanation

In C and C++ there are different modes (referred to as methods) of how floating point arithmetic is done, see (https://en.wikipedia.org/wiki/C99#IEEE_754_floating-point_support). You can detect which one is being used by the value of FLT_EVAL_METHOD found in cfloat. In mode 2 (which is what 32 bit g++ uses by default) all floating point arithmetic is done using long double. Note that in this mode numbers are temporarily stored as long doubles while being operated on, this can / will cause a kind of excess precision. In mode 0 (which is what 64 bit g++ uses by default) the arithmetic is done using each corresponding type, so there is no excess precision.

Detecting and turning on/off excess precision

Here is a simple example of how to detect excess precision (partly taken from https://stackoverflow.com/a/20870774)

// #pragma GCC target("fpmath=sse,sse2") // Turns off excess precision
// #pragma GCC target("fpmath=387") // Turns on excess precision

#include <iostream>
#include <cstdlib>
#include <cfloat>
using namespace std;

int main() {
  cout << "This is compiled in mode "<< FLT_EVAL_METHOD << '\n';
  cout << "0 means no excess precision.\n";
  cout << "2 means there is excess precision.\n\n";
  
  cout << "The following test detects excess precision\n";
  cout << "0 if no excess precision, or 8e-17 if there is excess precision.\n";
  double a = atof("1.2345678");
  double b = a*a;
  cout << b - 1.52415765279683990130 << '\n';
  return 0;
}

If b is rounded (as one would "expect" since it is a double), then the result is zero. Otherwise it is something like 8e-17 because of excess precision. I tried running this in custom invocation. MSVC(C++17), Clang and g++17(64bit) all use mode 0 and round b to 0, while g++11, g++14 and g++17 as expected all use mode 2 and b = 8e-17.

The culprit behind all of this misery is the old x87 instruction set, which only supports (80 bit) long double arithmetic. The modern solution is to on top of this use the SSE instruction set (version 2 or later), which supports both float and double arithmetic. On GCC you can turn this on with the flags -mfpmath=sse -msse2. This will not change the value of FLT_EVAL_METHOD, but it will effectively turn off excess precision, see 81993714.

It is also possible to effectively turn on excess precision with -mfpmath=387, see 81993724.

Fun exercise

Using your newfound knowledge of excess precision, try to find a compiler + input to "hack" this

#include <iostream>
#include <cmath>
using namespace std;

bool f() {
    double x;
    cin >> x;
    
    double y = x + 1.0;
    if (y >= 1.0)
        return false;
    
    int w = pow(y, 2);
    
    if (y < 1.0)
        return false;
    return y == 1.0;
}

int main() {
    if (f())
        cout << "HACKED\n";
    else
        cout << "Not hacked\n";
}

Conclusion / TLDR

32 bit g++ by default does all of its floating point arithmetic with (80 bit) long double. This causes a ton of frustrating and weird behaviours. 64 bit g++ does not have this issue.