Dividing two integers and rounding up the result, without using floating point

Benchmarks

Since a lot of different methods are shown in the answers and none of the answers actually prove any advantages in terms of performance I tried to benchmark them myself. My plan was to write an answer that contains a short table and a definite answer which method is the fastest.

Unfortunately it wasn't that simple. (It never is.) It seems that the performance of the rounding formulas depend on the used data type, compiler and optimization level.

In one case there is an increase of speed by 7.5× from one method to another. So the impact can be significant for some people.

TL;DR

For long integers the naive version using a type cast to float and std::ceil was actually the fastest. This was interesting for me personally since I intended to use it with size_t which is usually defined as unsigned long.

For ordinary ints it depends on your optimization level. For lower levels @Jwodder's solution performs best. For the highest levels std::ceil was the optimal one. With one exception: For the clang/unsigned int combination @Jwodder's was still better.

The solutions from the accepted answer never really outperformed the other two. You should keep in mind however that @Jwodder's solution doesn't work with negatives.

Results are at the bottom.

Implementations

To recap here are the four methods I benchmarked and compared:

@Jwodder's version (Jwodder)

template<typename T>
inline T divCeilJwodder(const T& numerator, const T& denominator)
{
    return (numerator + denominator - 1) / denominator;
}

@Ben Voigt's version using modulo (VoigtModulo)

template<typename T>
inline T divCeilVoigtModulo(const T& numerator, const T& denominator)
{
    return numerator / denominator + (((numerator < 0) ^ (denominator > 0))
        && (numerator%denominator));
}

@Ben Voigt's version without using modulo (VoigtNoModulo)

inline T divCeilVoigtNoModulo(const T& numerator, const T& denominator)
{
    T truncated = numerator / denominator;
    return truncated + (((numerator < 0) ^ (denominator > 0))
        && (numerator - truncated*denominator));
}

OP's implementation (TypeCast)

template<typename T>
inline T divCeilTypeCast(const T& numerator, const T& denominator)
{
    return (int)std::ceil((double)numerator / denominator);
}

Methodology

In a single batch the division is performed 100 million times. Ten batches are calculated for each combination of Compiler/Optimization level, used data type and used implementation. The values shown below are the averages of all 10 batches in milliseconds. The errors that are given are standard deviations.

The whole source code that was used can be found here. Also you might find this script useful which compiles and executes the source with different compiler flags.

The whole benchmark was performed on a i7-7700K. The used compiler versions were GCC 10.2.0 and clang 11.0.1.

Results

Now without further ado here are the results:

Now I can finally get on with my life :P

With help from DyP, came up with the following branchless formula:

int idiv_ceil ( int numerator, int denominator )
{
    return numerator / denominator
             + (((numerator < 0) ^ (denominator > 0)) && (numerator%denominator));
}

It avoids floating-point conversions and passes a basic suite of unit tests, as shown here:

• http://ideone.com/3OrviU

---

Here's another version that avoids the modulo operator.

int idiv_ceil ( int numerator, int denominator )
{
    int truncated = numerator / denominator;
    return truncated + (((numerator < 0) ^ (denominator > 0)) &&
                                             (numerator - truncated*denominator));
}

• http://ideone.com/Z41G5q

The first one will be faster on processors where IDIV returns both quotient and remainder (and the compiler is smart enough to use that).

Assuming that both myIntNumber and myOtherInt are positive, you could do:

int myValue = (myIntNumber + myOtherInt - 1) / myOtherInt;

Maybe it is just easier to do a:

int result = dividend / divisor;
if(dividend % divisor != 0)
    result++;