Cosine rounding failure on macOS

[zfergus]

I am running into issues with the interval library when compiling with Clang. This is the simple failure case:

#include <iomanip>
#include <iostream>

#include <boost/numeric/interval.hpp>

typedef boost::numeric::interval<
    double,
    boost::numeric::interval_lib::policies<
        boost::numeric::interval_lib::save_state<
            boost::numeric::interval_lib::rounded_transc_std<double> >,
        boost::numeric::interval_lib::checking_base<double> > >
    Interval;

int main(int argc, char* argv[])
{
    Interval theta = Interval(0.79358805865013693);
    Interval output = cos(theta);
    std::cout << std::setprecision(std::numeric_limits<double>::digits10 + 1)
              << "[" << output.lower() << ", " << output.upper() << "]"
              << std::endl;
    std::cout << "is empty: "
              << (output.lower() > output.upper() ? "true" : "false")
              << std::endl;
}

The output when compiled with clang is:

[0.7012920012119436, 0.7012920012119435]
is empty: true

You can see that the last digit was rounded improperly resulting in an empty interval. If I try the same case but use rounded_transc_exact for the rounding policy It works as expected:

[0.7012920012119437, 0.7012920012119437]
is empty: false

My compiler info is

Apple clang version 11.0.0 (clang-1100.0.33.17)
Target: x86_64-apple-darwin19.3.0
Thread model: posix

and the example is compiled in debug mode. The examples works fine on GCC (specifically I tested on gcc (Ubuntu 9.2.1-9ubuntu2) 9.2.1 20191008)

Is this an expected failure for Clang?

[zfergus]

Debugging some more, I wanted to verify it would work for GCC on my mac. Using g++-9 (Homebrew GCC 9.2.0_3) 9.2.0, I still get the same rounding failure:

[0.7012920012119436, 0.7012920012119435]
is empty: true

Could this be a problem with the hardware implementation of the trignometric functions? Is it guaranteed that FE_DOWNWARD cos(theta) < FE_UPWARD cos(theta)?

[zfergus]

At this point, I have run it on five different machines:

1. macOS with Intel(R) Core(TM) i7-7567U CPU: fail
2. Ubuntu 19.10 with AMD EPYC 7452 32-Core Processor: pass
3. Ubuntu 18.04.3 LTS with Intel(R) Core(TM) i7-5930K CPU: pass
4. CentOS Linux with Intel(R) Xeon(R) CPU E5-2680 v4: pass
5. CentOS Linux with Intel(R) Xeon(R) CPU E5-2690 v2: pass

All compiled with GCC (of different versions ranging from 7 to 9).

[jeking3]

Does this have any relation to #13?

[zfergus]

I don't think so, #13 was a compilation error if you did not specify the right rounding policy. This error is related to how the trigonometric functions are rounded using hardware rounding. We published a research article benchmarking various C/C++ interval arithmetic libraries. In it we found that the hardware rounding of transcendental functions is broken on certain platforms. We concluded filib to be the overall fastest and reliable library. You can read the full paper here.