BFloat16(::BigFloat) and BigFloat(::BFloat16)

[milankl]

We currently have

julia> BigFloat(BFloat16(1))
ERROR: MethodError: no method matching BigFloat(::BFloat16)

and

julia> convert(BFloat16,BigFloat(1))
ERROR: MethodError: no method matching BFloat16(::BigFloat)

This PR just defines those conversions via Float32, outside of the llvm_arithmetic blocks because I believe there's no LLVM functionality for BigFloat hence his needs to happen independently anyway.

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 66.66%. Comparing base (4b657c0) to head (e415c29).
Report is 23 commits behind head on master.

Additional details and impacted files

@@            Coverage Diff             @@
##           master      #63      +/-   ##
==========================================
+ Coverage   66.29%   66.66%   +0.36%     
==========================================
  Files           3        3              
  Lines         181      183       +2     
==========================================
+ Hits          120      122       +2     
  Misses         61       61              

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[maleadt]

This seems sensible, especially with bfloat being a truncated float, but I'm no floating-point expert so there may be an improved conversion. cc @oscardssmith maybe?

[milankl]

Maybe there's a way to do this faster, I don't know how Float32(::BigFloat) and the inverse are implemented. But Float32 is a superset of BFloat16 so you don't lose any precision if you convert to Float32 first.

[oscardssmith]

the BFloat16(::BigFloat) version of this will differ double rounding. I suggest looking at how the float16 conversation works in Julia

[milankl]

I see because of the cases where a-------x-b--------c, with a,c BFloat16s, b=(a+c)/2, a Float32 then x could be rounded to b::Float32 and subsequently to c even though it would round to a in a direct conversion

[milankl]

Wouldn't this happen here too?

BFloat16s.jl/src/bfloat16.jl

Lines 400 to 410 in ba5104d

	for F in (:abs, :abs2, :sqrt, :cbrt,
	:exp, :exp2, :exp10, :expm1,
	:log, :log2, :log10, :log1p,
	:sin, :cos, :tan, :csc, :sec, :cot,
	:asin, :acos, :atan, :acsc, :asec, :acot,
	:sinh, :cosh, :tanh, :csch, :sech, :coth,
	:asinh, :acosh, :atanh, :acsch, :asech, :acoth)
	@eval begin
	Base.$F(x::BFloat16) = BFloat16($F(Float32(x)))
	end
	end

[oscardssmith]

yes, but getting the elementary functions rounded correctly isn't a requirement (or easy), but correct rounding for arithmetic and conversion are relatively easy.

[ViralBShah]

Merge?

[oscardssmith]

as previously mentioned this version has double rounding, which can be avoided by using the same algorithm Float16 uses, but this is better than not having the capability

[maleadt]

Thanks; so let's merge this and open an issue to track the improvement.