Is `casecmp?` faster? -> No

bench.rb

require 'benchmark/ips'

Benchmark.ips do |x|
  x.report('upcase')   { 'String'.upcase == 'string' }
  x.report('downcase') { 'String'.downcase == 'string' }
  x.report('casecmp')  { 'String'.casecmp('string').zero? }
  x.report('casecmp?') { 'String'.casecmp?('string') }
  x.compare!
end

result.txt

Warming up --------------------------------------
              upcase   202.310k i/100ms
            downcase   209.096k i/100ms
             casecmp   225.472k i/100ms
            casecmp?   177.762k i/100ms
Calculating -------------------------------------
              upcase      3.841M (± 5.4%) i/s -     19.219M in   5.021148s
            downcase      3.860M (± 5.7%) i/s -     19.237M in   5.001815s
             casecmp      4.411M (± 6.6%) i/s -     22.096M in   5.035209s
            casecmp?      3.076M (± 3.2%) i/s -     15.465M in   5.033448s

Comparison:
             casecmp:  4410630.9 i/s
            downcase:  3860071.1 i/s - 1.14x  slower
              upcase:  3840503.1 i/s - 1.15x  slower
            casecmp?:  3075894.5 i/s - 1.43x  slower