Fix logabsbeta for BigFloat

[devmotion]

This PR adds a fallback definition for _loggammadiv. This fixes test errors in TuringLang/DistributionsAD.jl#172.

Motivation:
Version 1.4.1, or more concretely #316 (@andreasnoack), caused some regressions of logbeta and logabsbeta with BigFloats.

On 1.4.0:

julia> using SpecialFunctions

julia> beta(big(nextfloat(0.0)), big(nextfloat(8.0)))
2.024022533073106183524953467189173070495566497641421183569013580274303395679953e+323

julia> logbeta(big(nextfloat(0.0)), big(nextfloat(8.0)))
744.4400719213812623141072984460816341130871443029141429256103301959047499954517

julia> logabsbeta(big(nextfloat(0.0)), big(nextfloat(8.0)))
(744.4400719213812623141072984460816341130871443029141429256103301959047499954517, 1)

On 1.4.1:

julia> beta(big(nextfloat(0.0)), big(nextfloat(8.0)))
2.024022533073106183524953467189173070495566497641421183569013580274303395679953e+323

julia> logbeta(big(nextfloat(0.0)), big(nextfloat(8.0)))
ERROR: MethodError: no method matching _loggammadiv(::BigFloat, ::BigFloat)
Stacktrace:
 [1] loggammadiv(a::BigFloat, b::BigFloat)
   @ SpecialFunctions ~/.julia/packages/SpecialFunctions/W3J2O/src/beta_inc.jl:12
 [2] logabsbeta(a::BigFloat, b::BigFloat)
   @ SpecialFunctions ~/.julia/packages/SpecialFunctions/W3J2O/src/gamma.jl:822
 [3] logbeta(a::BigFloat, b::BigFloat)
   @ SpecialFunctions ~/.julia/packages/SpecialFunctions/W3J2O/src/gamma.jl:788
 [4] top-level scope
   @ REPL[15]:1

julia> logabsbeta(big(nextfloat(0.0)), big(nextfloat(8.0)))
ERROR: MethodError: no method matching _loggammadiv(::BigFloat, ::BigFloat)
Stacktrace:
 [1] loggammadiv(a::BigFloat, b::BigFloat)
   @ SpecialFunctions ~/.julia/packages/SpecialFunctions/W3J2O/src/beta_inc.jl:12
 [2] logabsbeta(a::BigFloat, b::BigFloat)
   @ SpecialFunctions ~/.julia/packages/SpecialFunctions/W3J2O/src/gamma.jl:822
 [3] top-level scope
   @ REPL[16]:1

beta still works since it uses the definition in mpfr on my machine. Actually, the problem was present in SpecialFunctions 1.4.0 as well but it is easier to trigger it in 1.4.1 since the branch is chosen not only for inputs with large difference in magnitude anymore (on 1.4.0:

SpecialFunctions.jl/src/gamma.jl

Line 842 in feccbf4

if abs(a) > 1e5*abs(b) && abs(a) > 1e5

, on 1.4.1:

SpecialFunctions.jl/src/gamma.jl

Line 821 in dd41481

if a > 0 && b > 8

).

On 1.4.0:

julia> logabsbeta(big(nextfloat(1e5)), big(1.0))
ERROR: MethodError: no method matching _loggammadiv(::BigFloat, ::BigFloat)
Stacktrace:
 [1] loggammadiv(a::BigFloat, b::BigFloat)
   @ SpecialFunctions ~/.julia/packages/SpecialFunctions/L13jJ/src/beta_inc.jl:12
 [2] logabsbeta(a::BigFloat, b::BigFloat)
   @ SpecialFunctions ~/.julia/packages/SpecialFunctions/L13jJ/src/gamma.jl:843
 [3] top-level scope
   @ REPL[14]:1

With this PR all examples work:

julia> beta(big(nextfloat(0.0)), big(nextfloat(8.0)))
2.024022533073106183524953467189173070495566497641421183569013580274303395679953e+323

julia> logbeta(big(nextfloat(0.0)), big(nextfloat(8.0)))
744.4400719213812623141072984460816341130871443029141429256103301959047499954517

julia> logabsbeta(big(nextfloat(0.0)), big(nextfloat(8.0)))
(744.4400719213812623141072984460816341130871443029141429256103301959047499954517, 1)

julia> logabsbeta(big(nextfloat(1e5)), big(1.0))
(-11.51292546497022856560910955709032851649991676439065368877238753498953004770512, 1)

[devmotion]

Test failures are the same as on the master branch, it seems.

[giordano]

Test failures are the same as on the master branch, it seems.

I'm lost, are they due to #319? Is there anything we can do to have a usable state?

[devmotion]

Hopefully #327 fixes it.

[codecov]

Codecov Report

Merging #325 (9e7461f) into master (f67c0f6) will decrease coverage by 0.03%.
The diff coverage is 100.00%.

@@            Coverage Diff             @@
##           master     #325      +/-   ##
==========================================
- Coverage   92.70%   92.66%   -0.04%     
==========================================
  Files          12       12              
  Lines        2645     2646       +1     
==========================================
  Hits         2452     2452              
- Misses        193      194       +1     

Flag	Coverage Δ
unittests	92.66% <100.00%> (-0.04%)	⬇️

Flags with carried forward coverage won't be shown. Click here to find out more.

Impacted Files	Coverage Δ
src/beta_inc.jl	90.54% <100.00%> (-0.30%)	⬇️
src/gamma.jl	94.91% <0.00%> (+0.26%)	⬆️

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[andreasnoack]

I think it would be useful to add a comment here as a reminder that a proper implementation should be added at some point.

[devmotion]

I added a comment.