<div dir="ltr"><div><div><div>In an effort to get this thread back on track, I tried implementing cos(_:) in pure generic Swift code, with the BinaryFloatingPoint protocol. It deviates from the _cos(_:) intrinsic by no more than 5.26362703423544e-11. Adding more terms to the approximation only has a small penalty to the performance for some reason.<br><br></div>To make the benchmarks fair, and explore the idea of distributing a Math module without killing people on the cross-module optimization boundary, I enabled some of the unsafe compiler attributes. All of these benchmarks are cross-module calls, as if the math module were downloaded as a dependency in the SPM.<br><br><span style="font-family:monospace,monospace">== Relative execution time (lower is better) ==<br><br></span><span style="font-family:monospace,monospace"><span style="background-color:rgb(217,210,233)">llvm intrinsic : 3.133</span><br></span></div><div><span style="font-family:monospace,monospace"><span style="background-color:rgb(234,209,220)">glibc cos() : 3.124</span><br></span></div><div><span style="font-family:monospace,monospace"><br>no attributes : 43.675<br>with specialization : 4.162<br>with inlining : 3.108<br>with inlining and specialization : 3.264</span><br><br></div>As you can see, the pure Swift generic implementation actually beats the compiler intrinsic (and the glibc cos() but I guess they’re the same thing) when inlining is used, but for some reason generic specialization and inlining don’t get along very well.<br><br></div>Here’s the source implementation. It uses a taylor series (!) which probably isn’t optimal but it does prove that cos() and sin() can be implemented as generics in pure Swift, be distributed as a module outside the stdlib, and still achieve competitive performance with the llvm intrinsics.<br><br><span style="font-family:monospace,monospace">@_inlineable<br>//@_specialize(where F == Float)<br>//@_specialize(where F == Double)<br>public<br>func cos<F>(_ x:F) -> F where F:BinaryFloatingPoint<br>{<br> let x:F = abs(x.remainder(dividingBy: 2 * F.pi)),<br> quadrant:Int = Int(x * (2 / F.pi))<br><br> switch quadrant<br> {<br> case 0:<br> return cos(on_first_quadrant: x)<br> case 1:<br> return -cos(on_first_quadrant: F.pi - x)<br> case 2:<br> return -cos(on_first_quadrant: x - F.pi)<br> case 3:<br> return -cos(on_first_quadrant: 2 * F.pi - x)<br> default:<br> fatalError("unreachable")<br> }<br>}<br><br>@_versioned<br>@_inlineable<br>//@_specialize(where F == Float)<br>//@_specialize(where F == Double)<br>func cos<F>(on_first_quadrant x:F) -> F where F:BinaryFloatingPoint<br>{<br> let x2:F = x * x<br> var y:F = -0.0000000000114707451267755432394<br> for c:F in [0.000000002087675698165412591559,<br> -0.000000275573192239332256421489,<br> 0.00002480158730158702330045157,<br> -0.00138888888888888880310186415,<br> 0.04166666666666666665319411988,<br> -0.4999999999999999999991637437,<br> 0.9999999999999999999999914771<br> ]<br> {<br> y = x2 * y + c<br> }<br> return y<br>}</span><br></div><div class="gmail_extra"><br><div class="gmail_quote">On Thu, Aug 3, 2017 at 7:04 AM, Stephen Canon via swift-evolution <span dir="ltr"><<a href="mailto:swift-evolution@swift.org" target="_blank">swift-evolution@swift.org</a>></span> wrote:<br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><div style="word-wrap:break-word;line-break:after-white-space"><span class=""><blockquote type="cite">On Aug 2, 2017, at 7:03 PM, Karl Wagner via swift-evolution <<a href="mailto:swift-evolution@swift.org" target="_blank">swift-evolution@swift.org</a>> wrote:<br></blockquote><div><blockquote type="cite"><div><br class="m_3712668855384799923Apple-interchange-newline"><span style="font-family:Helvetica;font-size:12px;font-style:normal;font-variant-caps:normal;font-weight:normal;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px;float:none;display:inline!important">It’s important to remember that computers are mathematical machines, and some functions which are implemented in hardware on essentially every platform (like sin/cos/etc) are definitely best implemented as compiler intrinsics.</span></div></blockquote><br></div></span><div>sin/cos/etc are implemented in software, not hardware. x86 does have the FSIN/FCOS instructions, but (almost) no one actually uses them to implement the sin( ) and cos( ) functions; they are a legacy curiosity, both too slow and too inaccurate for serious use today. There are no analogous instructions on ARM or PPC.</div><div><br></div><div>– Steve</div></div><br>______________________________<wbr>_________________<br>
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