Make compatible with FastDifferentiation 0.4.5

[duetosymmetry]

Related issues

Closes #110 (and thereby #120)

Checklist

• Tests are passing
• Lint workflow is passing
• Docs were updated and workflow is passing

FD.make_Expr changed so that in_place and init_with_zeros are both keyword args.

The code generation changed to have a bounds checking preamble before the
@inbounds (actual) code block.

The code generation changed to allow for multiple vector args, which means that
the first argument in the code is now named input_variables1 instead of
input_variables.

Minor bugfixes in ArgumentError strings that tried to interpolate the old
variable name $pnsystem instead of $fdpnsystem.

[duetosymmetry]

Sorry for the force push, I didn't realize the linter ran JuliaFormatter and required no changes. Just updated so that JuliaFormatter will be happy.

[duetosymmetry]

@moble any idea why the test is failing? It's in the test of the Hessian of the "full-order" system. Snippet:

using PostNewtonian
using ForwardDiff
using Random

Random.seed!(1234)
pn = rand(PostNewtonian.BBH; v=0.5)
function c5(u)
    pnsystem = PostNewtonian.BBH(u; PNOrder=PostNewtonian.pn_order(pn))
    inspiral = PostNewtonian.orbital_evolution(pnsystem; vₑ=0.6)
    h = PostNewtonian.inertial_waveform(inspiral)
    return sum(abs2, h)
end

∇c = ForwardDiff.gradient(c5, pn.state)
# Yields: 
# 14-element Vector{Float64}:
#   14.990245312394162
#  -54.357556232368374
#   -6.687492593541796
#   -7.527668111301677
#   11.21251862573012
#   -1.5455683131206928
#   -1.8142250030527391
#    2.3620067547668357
#   -7.476167782324888
#    0.700624861014795
#    4.86046429300934
#  -11.094029185528715
#   54.673132167863805
#    0.0

Hc = ForwardDiff.hessian(c5, pn.state)
# Yields:
# 14×14 Matrix{Float64}:
#    399.614    -1389.0     -132.137    -145.431     225.626    -27.6861    -30.1599      65.7419    -220.586     21.8922     145.274    -337.84      700.567    0.0
#  -1365.22      4794.07     434.865     480.858    -742.702     92.5739    101.411     -222.673      739.545    -72.0894    -486.346    1129.0     -2658.34     0.0
#   -130.451      436.316     41.1711     44.8527    -67.6969     9.54944    10.4588     -22.2959      52.4884   -19.0623     -48.7998    128.062    -208.203    0.0
#   -142.111      477.269     44.3957     47.6051    -74.9423    10.2845     10.9258     -26.8847      67.055     -6.86303    -44.012     101.579    -286.812    0.0
#    222.329     -743.632    -67.5615    -75.5699    117.094    -15.6639    -17.5785      43.7195    -158.367      5.54988     93.6726   -206.151     478.234    0.0
#    -27.3708      93.1871     9.57365    10.4177    -15.7366     2.25412     2.38725    -28.6024      85.9849   -12.1013     -59.469     140.17     -546.023    0.0
#    -29.3763     100.479     10.345      10.9141    -17.4187     2.35469     2.50555     -5.56245     13.5658    -1.41221     -8.9086     20.6221    -52.1816   0.0
#    -23.9911      70.1501     4.63845     3.36708    -6.86618    1.2446      0.64836      5.61851    -21.7463     0.155569    12.3307    -26.7754     33.9043   0.0
#     96.3558    -294.987    -42.9019    -39.1863     17.4661   -10.3298     -8.45636    -21.7134      57.8518    -3.15036    -24.0939     89.9042   -109.216    0.0
#     51.7415    -169.663    -27.8404    -15.0235     17.9871    -6.65938    -3.42806      0.188049    -3.26294    9.41366      9.83746   -17.1024      5.81976  0.0
#    183.595     -612.154    -58.6041    -49.5317     99.3879   -13.7712    -11.328       12.2971     -24.0556     9.7603      15.6868    -49.2973     69.3304   0.0
#   -402.228     1340.72     144.049     107.008    -207.355     33.6163     24.556      -26.6963      89.8102   -16.9245     -49.2934    114.939    -168.615    0.0
#    756.036    -2850.62    -208.599    -241.247     366.503    -46.2126    -54.1972      36.744     -118.839      6.03694     75.8516   -183.64      957.119    0.0
#      7.14202    -23.3337    -2.08946    -2.18007     3.51506   -0.493543   -0.489158     0.0          0.0        0.0          0.0         0.0         0.0      0.0

So, obviously the last row is not all 0 like the test requires. But the Hessian isn't even symmetric... I don't know how this happened.