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

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

"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.

The issue is caused by something called excess precision. In C and C++ there are multiple 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 float.h. In mode 2 (which is what 32 bit g++ uses by default) all floating point arithmetic is done using long double. 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.

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

#include <stdio.h>
#include <stdlib.h>
#include <float.h>

int main() {
  printf("This is compiled in mode %d\n", FLT_EVAL_METHOD);
  printf("0 means no excess precision.\n");
  printf("2 means it has excess precision.\n\n");
  
  printf("The following test detects excess precision\n");
  printf("0 if no excess precision, or 8e-17 if excess precision is used.\n");
  double a = atof("1.2345678");
  double b = a*a;
  printf("%.20e\n", b - 1.52415765279683990130);
  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, 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 is 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 (version 2 or later) instruction set, which supports both float and double arithmetic. GCC got a flag to turn this on -mfpmath=sse -msse2. This will not change the value of FLT_EVAL_METHOD, but it will effectively turn off excess precision, see 81988982.

It is also possible to turn on excess precision with -mfpmath=387, see 81989001.