exp(pi*i) produces slightly incorrect result

[mgeck64]

Basically, exp(pi*i) where i is the imaginary unit should produce -1 but produces a slightly different result with Boost 1.77.0; however it produces the correct result with Boost 1.74.0. The following program reproduces the issue:

#include <boost/multiprecision/cpp_complex.hpp>

using CT = boost::multiprecision::cpp_complex_50;
static const auto pi = boost::math::constants::pi<CT::value_type>();
static const auto i  = CT(0, 1);

int main() {
    std::cout << exp(pi*i) << std::endl;
    // Output when compiled with Boost 1.74.0 is -1 as expected.
    // Output when compiled with Boost 1.77.0 is (-1,4.33483e-51).
    // Note: The equivalent program using std::complex when run in compiler
    // explorer produces a similar result: (-1,1.22465e-16). Don't know if this
    // could be related.
    // Note 2: My version of g++ under which this was compiled is 11.2.0.
}

[NAThompson]

Since π is not representable, we must apply the rounding model of floating point arithmetic: π̂=π(1+μ), where μ is the unit roundoff.

Then:

exp(iπ̂) = exp(iπ)exp(iμπ) = -cos(μπ) -isin(μπ) ≈ -1 -iμπ,

which is exactly as you observe.

[mgeck64]

But why would version 1.74.0 produce -1 (the desired result) but 1.77.0 produce (-1,4.33483e-51)?

On Wed, Nov 17, 2021 at 10:12 pm, Mark Geck @.***> wrote:

On Wed, Nov 17, 2021 at 8:53 pm, Nick @.***> wrote:

Since π is not representable, we must apply the rounding model of floating point arithmetic: π̂=π(1+μ), where μ is the unit roundoff.

Then:

exp(iπ̂) = exp(iπ)exp(iμπ) = -isin(μπ) ≈ -iμπ,

which is exactly as you observe.

— You are receiving this because you authored the thread. Reply to this email directly, view it on GitHub https://github.com/boostorg/multiprecision/issues/392#issuecomment-972537176, or unsubscribe https://github.com/notifications/unsubscribe-auth/ARELBE7QBM7GXT2GVTW2R2LUMSBE3ANCNFSM5IIUZC5A. Triage notifications on the go with GitHub Mobile for iOS https://apps.apple.com/app/apple-store/id1477376905?ct=notification-email&mt=8&pt=524675 or Android https://play.google.com/store/apps/details?id=com.github.android&referrer=utm_campaign%3Dnotification-email%26utm_medium%3Demail%26utm_source%3Dgithub.

[mgeck64]

More precisely stated, my question is why sin(pi) correctly evaluates to 0 in version 1.74 but not in 1.77. Note also 1.75 produces the same result as 1.77 so whatever changed happened between 1.74 and 1.75.

Note also that google calculator and wolfram alpha evaluate sin(pi) to 0. In particular, wolfram alpha will evaluate sin(3.1415926535897932384626433832795028841971693993751) to 0.

Is pi being treated specially in 1.74 and the other calculators?

[ckormanyos]

I made some investigations to give us some content for the discussion.

We are dealing with finite-precision numerical representations and should discuss what makes sense for really small numbers.

#include <boost/multiprecision/cpp_complex.hpp>

using CT = boost::multiprecision::cpp_complex_50;
using component_value_type = typename CT::value_type;
static const auto pi = boost::math::constants::pi<component_value_type>();
static const auto i  = CT(0, 1);

int main()
{
  {
    std::stringstream strm;

    strm << std::setprecision(std::numeric_limits<component_value_type>::digits10)
         << exp(pi*i)
         << std::endl;

    std::cout << strm.str() << std::endl;
  }

  {
    std::stringstream strm;

    strm << std::fixed
         << std::setprecision(std::numeric_limits<component_value_type>::digits10)
         << exp(pi*i)
         << std::endl;

    std::cout << strm.str() << std::endl;
  }

  {
    std::stringstream strm;

    strm << std::fixed
         << exp(pi*i)
         << std::endl;

    std::cout << strm.str() << std::endl;
  }
}

[jzmaddock]

I believe this is a result of addressing this issue: https://github.com/boostorg/multiprecision/issues/264 via: https://github.com/boostorg/multiprecision/pull/270

Which means we now do extended precision argument reduction, so reducing an input of PI, no longer results in 0 but something very close to it. I'm in two minds what to do here.