Compare floating point numbers as integers

Question

Can two floating point values (IEEE 754 binary64) be compared as integers? Eg.

long long a = * (long long *) ptr_to_double1,
          b = * (long long *) ptr_to_double2;
if (a < b) {...}

assuming the size of long long and double is the same.

Answer 1

YES - Comparing the bit-patterns for two floats as if they were integers (aka "type-punning") produces meaningful results under some restricted scenarios...

Identical to floating-point comparison when:

• Both numbers are positive, positive-zero, or positive-infinity.
• One positive and one negative number, and you are using a signed integer comparison.

Inverse of floating-point comparison when:

• Both numbers are negative, negative-zero, or negative-infinity.
• One positive and one negative number, and you are using a unsigned integer comparison.

Not comparable to floating-point comparison when:

• Either number is one of the NaN values - Floating point comparisons with a NaN always returns false, and this simply can't be modeled in integer operations where exactly one of the following is always true: (A < B), (A == B), (B < A).

Negative floating-point numbers are a bit funky b/c they are handled very differently than in the 2's complement arithmetic used for integers. Doing an integer +1 on the representation for a negative float will make it a bigger negative number.

With a little bit manipulation, you can make both positive and negative floats comparable with integer operations (this can come in handy for some optimizations):

int32 float_to_comparable_integer(float f) {
  uint32 bits = std::bit_cast<uint32>(f);
  const uint32 sign_bit = bits & 0x80000000ul;
  // Modern compilers turn this IF-statement into a conditional move (CMOV) on x86,
  // which is much faster than a branch that the cpu might mis-predict.
  if (sign_bit) {
    bits = 0x7FFFFFF - bits;
  }
  return static_cast<int32>(bits);
}

Again, this does not work for NaN values, which always return false from comparisons, and have multiple valid bit representations:

• Signaling NaNs (w/ sign bit): Anything between 0xFF800001, and 0xFFBFFFFF.
• Signaling NaNs (w/o sign bit): Anything between 0x7F800001, and 0x7FBFFFFF.
• Quiet NaNs (w/ sign bit): Anything between 0xFFC00000, and 0xFFFFFFFF.
• Quiet NaNs (w/o sign bit): Anything between 0x7FC00000, and 0x7FFFFFFF.

IEEE-754 bit format: http://www.puntoflotante.net/FLOATING-POINT-FORMAT-IEEE-754.htm

More on Type-Punning: https://randomascii.wordpress.com/2012/01/23/stupid-float-tricks-2/

Answer 2

No. Two floating point values (IEEE 754 binary64) cannot compare simply as integers with if (a < b).

IEEE 754 binary64

The order of the values of double is not the same order as integers (unless you are are on a rare sign-magnitude machine). Think positive vs. negative numbers.

double has values like 0.0 and -0.0 which have the same value but different bit patterns.

double has "Not-a-number"s that do not compare like their binary equivalent integer representation.

If both the double values were x > 0 and not "Not-a-number", endian, aliasing, and alignment, etc. were not an issue, OP's idea would work.

Alternatively, a more complex if() ... condition would work - see below

[non-IEEE 754 binary64]

Some double use an encoding where there are multiple representations of the same value. This would differ from an "integer" compare.

---

Tested code: needs 2's complement, same endian for double and the integers, does not account for NaN.

int compare(double a, double b) {
  union {
    double d;
    int64_t i64;
    uint64_t u64;
  } ua, ub;
  ua.d = a;
  ub.d = b;
  // Cope with -0.0 right away
  if (ua.u64 == 0x8000000000000000) ua.u64 = 0;
  if (ub.u64 == 0x8000000000000000) ub.u64 = 0;
  // Signs differ?
  if ((ua.i64 < 0) != (ub.i64 < 0)) {
    return ua.i64 >= 0 ? 1 : -1;
  }
  // If numbers are negative
  if (ua.i64 < 0) {
    ua.u64 = -ua.u64;
    ub.u64 = -ub.u64;
  }
  return (ua.u64 > ub.u64)  - (ua.u64 < ub.u64);
}

Thanks to @David C. Rankin for a correction.

Test code

void testcmp(double a, double b) {
  int t1 = (a > b) - (a < b);
  int t2 = compare(a, b);
  if (t1 != t2) {
    printf("%le %le %d %d\n", a, b, t1, t2);
  }

}

#include <float.h>
void testcmps() {
  // Various interesting `double`
  static const double a[] = { 
      -1.0 / 0.0, -DBL_MAX, -1.0, -DBL_MIN, -0.0, 
      +0.0, DBL_MIN, 1.0, DBL_MAX, +1.0 / 0.0 };

  int n = sizeof a / sizeof a[0];
  for (int i = 0; i < n; i++) {
    for (int j = 0; j < n; j++) {
      testcmp(a[i], a[j]);
    }
  }
  puts("!");
}

Answer 3

If you strictly cast the bit value of a floating point number to its correspondingly-sized signed integer (as you've done), then signed integer comparison of the results will be identical to the comparison of the original floating-point values, excluding NaN values. Put another way, this comparison is legitimate for all representable finite and infinite numeric values.

In other words, for double-precision (64-bits), this comparison will be valid if the following tests pass:

long long exponentMask = 0x7ff0000000000000;
long long mantissaMask = 0x000fffffffffffff;

bool isNumber =  ((x & exponentMask) != exponentMask)  // Not exp 0x7ff
              || ((x & mantissaMask) == 0);            // Infinities

for each operand x.

Of course, if you can pre-qualify your floating-point values, then a quick isNaN() test would be much more clear. You'd have to profile to understand performance implications.

Answer 4

There are two parts to your question:

1. Can two floating point numbers be compared? The answer to this is yes. it is perfectly valid to compare size of floating point numbers. Generally you want to avoid equals comparisons due to truncation issues see here, but

   if (a < b)
   

   will work just fine.

2. Can two floating point numbers be compared as integers? This answer is also yes, but this will require casting. This question should help with that answer: convert from long long to int and the other way back in c++