> On 3. Aug 2017, at 20:52, Taylor Swift via swift-evolution
> <swift-evolution@swift.org> wrote:
>
> 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.
>
> 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.
>
> == Relative execution time (lower is better) ==
>
> llvm intrinsic : 3.133
> glibc cos() : 3.124
>
> no attributes : 43.675
> with specialization : 4.162
> with inlining : 3.108
> with inlining and specialization : 3.264
>
> 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.
>
> 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.
>
> @_inlineable
> //@_specialize(where F == Float)
> //@_specialize(where F == Double)
> public
> func cos<F>(_ x:F) -> F where F:BinaryFloatingPoint
> {
> let x:F = abs(x.remainder(dividingBy: 2 * F.pi)),
> quadrant:Int = Int(x * (2 / F.pi))
>
> switch quadrant
> {
> case 0:
> return cos(on_first_quadrant: x)
> case 1:
> return -cos(on_first_quadrant: F.pi - x)
> case 2:
> return -cos(on_first_quadrant: x - F.pi)
> case 3:
> return -cos(on_first_quadrant: 2 * F.pi - x)
> default:
> fatalError("unreachable")
> }
> }
>
> @_versioned
> @_inlineable
> //@_specialize(where F == Float)
> //@_specialize(where F == Double)
> func cos<F>(on_first_quadrant x:F) -> F where F:BinaryFloatingPoint
> {
> let x2:F = x * x
> var y:F = -0.0000000000114707451267755432394
> for c:F in [0.000000002087675698165412591559,
> -0.000000275573192239332256421489,
> 0.00002480158730158702330045157,
> -0.00138888888888888880310186415,
> 0.04166666666666666665319411988,
> -0.4999999999999999999991637437,
> 0.9999999999999999999999914771
> ]
> {
> y = x2 * y + c
> }
> return y
> }
>
> On Thu, Aug 3, 2017 at 7:04 AM, Stephen Canon via swift-evolution
> <swift-evolution@swift.org <mailto:swift-evolution@swift.org>> wrote:
>> On Aug 2, 2017, at 7:03 PM, Karl Wagner via swift-evolution
>> <swift-evolution@swift.org <mailto:swift-evolution@swift.org>> wrote:
>>
>> 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.
>
> 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.
>
> – Steve
>
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Just a guess, but I’d expect inlining implies specialisation. It would be weird
if the compiler inlined a chunk of unoptimised generic code in to your function.
Pretty cool figures, though.
- Karl
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