How to test for lossless double / integer conversi

问题:

I have one double, and one int64_t. I want to know if they hold exactly the same value, and if converting one type into the other does not lose any information.

My current implementation is the following:

int int64EqualsDouble(int64_t i, double d) {
    return (d >= INT64_MIN)
        && (d < INT64_MAX)
        && (round(d) == d)
        && (i == (int64_t)d);
}

My question is: is this implementation correct? And if not, what would be a correct answer? To be correct, it must leave no false positive, and no false negative.

Some sample inputs:

• int64EqualsDouble(0, 0.0) should return 1
• int64EqualsDouble(1, 1.0) should return 1
• int64EqualsDouble(0x3FFFFFFFFFFFFFFF, (double)0x3FFFFFFFFFFFFFFF) should return 0, because 2^62 - 1 can be exactly represented with int64_t, but not with double.
• int64EqualsDouble(0x4000000000000000, (double)0x4000000000000000) should return 1, because 2^62 can be exactly represented in both int64_t and double.
• int64EqualsDouble(INT64_MAX, (double)INT64_MAX) should return 0, because INT64_MAX can not be exactly represented as a double
• int64EqualsDouble(..., 1.0e100) should return 0, because 1.0e100 can not be exactly represented as an int64_t.

回答1:

Yes, your solution works correctly because it was designed to do so, because int64_t is represented in two's complement by definition (C99 7.18.1.1:1), on platforms that use something resembling binary IEEE 754 double-precision for the double type. It is basically the same as this one.

Under these conditions:

• d < INT64_MAX is correct because it is equivalent to d < (double) INT64_MAX and in the conversion to double, the number INT64_MAX, equal to 0x7fffffffffffffff, rounds up. Thus you want d to be strictly less than the resulting double to avoid triggering UB when executing (int64_t)d.

• On the other hand, INT64_MIN, being -0x8000000000000000, is exactly representable, meaning that a double that is equal to (double)INT64_MIN can be equal to some int64_t and should not be excluded (and such a double can be converted to int64_t without triggering undefined behavior)

It goes without saying that since we have specifically used the assumptions about 2's complement for integers and binary floating-point, the correctness of the code is not guaranteed by this reasoning on platforms that differ. Take a platform with binary 64-bit floating-point and a 64-bit 1's complement integer type T. On that platform T_MIN is -0x7fffffffffffffff. The conversion to double of that number rounds down, resulting in -0x1.0p63. On that platform, using your program as it is written, using -0x1.0p63 for d makes the first three conditions true, resulting in undefined behavior in (T)d, because overflow in the conversion from integer to floating-point is undefined behavior.

---

If you have access to full IEEE 754 features, there is a shorter solution:

#include <fenv.h>
…
#pragma STDC FENV_ACCESS ON
feclearexcept(FE_INEXACT), f == i && !fetestexcept(FE_INEXACT)

This solution takes advantage of the conversion from integer to floating-point setting the INEXACT flag iff the conversion is inexact (that is, if i is not representable exactly as a double).

The INEXACT flag remains unset and f is equal to (double)i if and only if f and i represent the same mathematical value in their respective types.

This approach requires the compiler to have been warned that the code accesses the FPU's state, normally with #pragma STDC FENV_ACCESS on but that's typically not supported and you have to use a compilation flag instead.

回答2:

OP's code has a dependency that can be avoided.

For a successful compare, d must be a whole number and round(d) == d takes care of that. Even d, as a NaN would fail that.

d must be mathematically in the range of [INT64_MIN ... INT64_MAX] and if the if conditions properly insure that, then the final i == (int64_t)d completes the test.

So the question comes down to comparing INT64 limits with the double d.

Let us assume FLT_RADIX == 2, but not necessarily IEEE 754 binary64.

d >= INT64_MIN is not a problem as -INT64_MIN is a power of 2 and exactly converts to a double of the same value, so the >= is exact.

Code would like to do the mathematical d <= INT64_MAX, but that may not work and so a problem. INT64_MAX is a "power of 2 - 1" and may not convert exactly - it depends on if the precision of the double exceeds 63 bits - rendering the compare unclear. A solution is to halve the comparison. d/2 suffers no precision loss and INT64_MAX/2 + 1 converts exactly to a double power-of-2

d/2 < (INT64_MAX/2 + 1)

[Edit]

// or simply
d < ((double)(INT64_MAX/2 + 1))*2

---

Thus if code does not want to rely on the double having less precision than uint64_t. (Something that likely applies with long double) a more portable solution would be

int int64EqualsDouble(int64_t i, double d) {
    return (d >= INT64_MIN)
        && (d < ((double)(INT64_MAX/2 + 1))*2)  // (d/2 < (INT64_MAX/2 + 1))
        && (round(d) == d)
        && (i == (int64_t)d);
}

Note: No rounding mode issues.

[Edit] Deeper limit explanation

Insuring mathematically, INT64_MIN <= d <= INT64_MAX, can be re-stated as INT64_MIN <= d < (INT64_MAX + 1) as we are dealing with whole numbers. Since the raw application of (double) (INT64_MAX + 1) in code is certainly 0, an alternative, is ((double)(INT64_MAX/2 + 1))*2. This can be extended for rare machines with double of higher powers-of-2 to ((double)(INT64_MAX/FLT_RADIX + 1))*FLT_RADIX. The comparison limits being exact powers-of-2, conversion to double suffers no precision loss and (lo_limit >= d) && (d < hi_limit) is exact, regardless of the precision of the floating point. Note: that a rare floating point with FLT_RADIX == 10 is still a problem.

回答3:

In addition to Pascal Cuoq's elaborate answer, and given the extra context you give in comments, I would add a test for negative zeros. You should preserve negative zeros unless you have good reasons not to. You need a specific test to avoid converting them to (int64_t)0. With your current proposal, negative zeros will pass your test, get stored as int64_t and read back as positive zeros.

I am not sure what is the most efficient way to test them, maybe this:

int int64EqualsDouble(int64_t i, double d) {
    return (d >= INT64_MIN)
        && (d < INT64_MAX)
        && (round(d) == d)
        && (i == (int64_t)d
        && (!signbit(d) || d != 0.0);
}