benchmarks used to compare the PR for `rational.jl` with the current `rational.jl`

BenchmarkingRationals.jl

using LinearAlgebra

#=
    you set this directory with the const or through the environment
    ensure RelativeSpeeds.jl and Fractions.jl are kept there
=#
const DefaultBenchmarkingDir = pwd()
benchmarking_dir = haskey(ENV, "BenchmarkingDir") ? ENV["BenchmarkingDir"] : DefaultBenchmarkingDir

cd(benchmarking_dir)
include("RelativeSpeeds.jl")
include("Fractions.jl")
using .Fractions

#=
    the first 256 composite5 numbers
    2*3*5*7*11 and 2^3 * 3^3 * 5^1 * 7^1 * 19^1 are examples
=#
const composite5 = [
    2310, 2730, 3570, 3990, 4290, 4620, 4830, 5460, 5610, 6006, 6090, 6270,
    6510, 6630, 6930, 7140, 7410, 7590, 7770, 7854, 7980, 8190, 8580, 8610,
    8778, 8970, 9030, 9240, 9282, 9570, 9660, 9690, 9870, 10010, 10230,
    10374, 10626, 10710, 10920, 11130, 11220, 11310, 11550, 11730, 11970,
    12012, 12090, 12180, 12210, 12390, 12540, 12558, 12810, 12870, 13020,
    13090, 13110, 13260, 13398, 13530, 13566, 13650, 13860, 14070, 14190,
    14280, 14322, 14430, 14490, 14586, 14630, 14790, 14820, 14910, 15015,
    15180, 15330, 15470, 15510, 15540, 15708, 15810, 15834, 15960, 15990,
    16170, 16302, 16380, 16422, 16530, 16590, 16770, 16830, 16926, 17094,
    17160, 17220, 17290, 17430, 17490, 17556, 17670, 17710, 17850, 17940,
    18018, 18060, 18270, 18330, 18354, 18480, 18564, 18690, 18810, 18870,
    18942, 19110, 19140, 19320, 19380, 19470, 19530, 19635, 19734, 19740,
    19866, 19890, 19950, 20010, 20020, 20130, 20202, 20370, 20460, 20670,
    20706, 20748, 20790, 20910, 20930, 21090, 21210, 21252, 21318, 21390,
    21420, 21450, 21630, 21714, 21840, 21930, 21945, 22110, 22134, 22230,
    22260, 22330, 22386, 22440, 22470, 22610, 22620, 22770, 22890, 23010,
    23100, 23142, 23205, 23310, 23370, 23430, 23460, 23478, 23562, 23730,
    23790, 23870, 23940, 23970, 24024, 24090, 24150, 24180, 24310, 24360,
    24420, 24486, 24510, 24570, 24738, 24780, 24882, 24990, 25080, 25116,
    25194, 25410, 25530, 25620, 25662, 25740, 25806, 25830, 25935, 26040,
    26070, 26130, 26180, 26220, 26334, 26390, 26418, 26520, 26565, 26598,
    26670, 26790, 26796, 26910, 26970, 27030, 27060, 27090, 27132, 27170,
    27258, 27300, 27370, 27390, 27510, 27690, 27720, 27846, 27930, 28014,
    28050, 28140, 28182, 28210, 28290, 28380, 28470, 28490, 28560, 28644,
    28710, 28770, 28842, 28860, 28938, 28980, 29070, 29172, 29190, 29260,
    29274
 ];

const rationals256 = composite5 .// reverse(composite5);
const fractions256 = composite5 .⨸ reverse(composite5);

const rationals64 = rationals256[96:159]
const fractions64 = fractions256[96:159]
const rationals16 = rationals256[120:135]
const fractions16 = fractions256[120:135]
const rationals4 = rationals256[126:129]
const fractions4 = fractions256[126:129]
const rationals2 = rationals256[127:2:129]
const fractions2 = fractions256[127:2:129]
const rationals1 = rationals256[64]
const fractions1 = fractions256[64]

const rationals16x16 = Rational{BigInt}.(reshape(rationals256, 16, 16))
const fractions16x16 = Rational{BigInt}.(reshape(fractions256, 16, 16))
const rationals8x8   = Rational{BigInt}.(reshape(rationals64, 8, 8))
const fractions8x8   = Rational{BigInt}.(reshape(fractions64, 8, 8))
const rationals4x4   = Rational{Int128}.(reshape(rationals16, 4, 4))
const fractions4x4   = Rational{Int128}.(reshape(fractions16, 4, 4))
const rationals2x2   = Rational{Int64}.(reshape(rationals4, 2, 2))
const fractions2x2   = Rational{Int64}.(reshape(fractions4, 2, 2))

#=
    relspeed

relspeed | interpretation
---------|---------------
1.0      | no change in performance
1.5      | what took 3 time units now takes 2 (3/2 == 1.5)
2.0      | what took 2 time units now takes 1
2.5      | what took 5 time units now takes 2

=#
relspeed(x->x[1] + x[2], rationals2, fractions2)    # 1.0
relspeed(x->x[1] - x[2], rationals2, fractions2)    # 1.0
relspeed(x->x[1] * x[2], rationals2, fractions2)    # 1.6
relspeed(x->x[1] / x[2], rationals2, fractions2)    # 1.55
relspeed(inv, rationals1, fractions1)               # 2.8

relspeed(prod, rationals4, fractions4)                                             # 1.5
relspeed(prod, rationals16, fractions16)                                           # 1.6
relspeed(prod, Rational{Int128}.(rationals64), Fraction{Int128}.(fractions64))     # 1.4
relspeed(prod, Rational{BigInt}.(rationals256), Fraction{BigInt}.(fractions256))   # 1.4

relspeed(x->inv.(x), rationals4, fractions4)                                       # 2.1
relspeed(x->inv.(x), rationals16, fractions16)                                     # 2.4
relspeed(x->inv.(x), rationals64, fractions64)                                     # 2.6
relspeed(x->inv.(x), rationals256, fractions256)                                   # 2.8

relspeed(dot, rationals4, rationals4, fractions4, fractions4)                      # 1.2
relspeed(dot, Rational{BigInt}.(rationals16), Rational{BigInt}.(rationals16),
              Fraction{BigInt}.(fractions16), Fraction{BigInt}.(fractions16) )     # 1.0


relspeed(*, rationals16x16, rationals16x16, fractions16x16, fractions16x16)        # 1.0
relspeed(+, rationals4x4, rationals4x4, fractions4x4, fractions4x4)                # 1.0

relspeed(inv, rationals4x4, fractions4x4)                                          # 1.0
relspeed(inv, rationals2x2, fractions2x2)                                          # 1.0