[Bug c/81630] New: powl returns values with insufficient accuracy

["kakehi at waseda dot jp" <gcc-bugzilla at gcc dot gnu dot org>]

https://gcc.gnu.org/bugzilla/show_bug.cgi?id=81630

            Bug ID: 81630
           Summary: powl returns values with insufficient accuracy
           Product: gcc
           Version: 8.0
            Status: UNCONFIRMED
          Severity: normal
          Priority: P3
         Component: c
          Assignee: unassigned at gcc dot gnu.org
          Reporter: kakehi at waseda dot jp
  Target Milestone: ---

Created attachment 41870
  --> https://gcc.gnu.org/bugzilla/attachment.cgi?id=41870&action=edit
compare powl(t,N) with exp2l(N*log2l(t))

I calculated powl(t,N) where t= (N+1)/N, N: an integer value 2^n (n: positive
integer) to get an approximation to Napier's constant.
Here is some results:
  n          powl(t, N)                        exp2l(N*log2l(t))
 25: 2.7182817879534639155837683e+00  2.7182817879534906457209692e+00
 26: 2.7182818082062409458920377e+00  2.7182818082062676638861742e+00
 27: 2.7182818183325918449770586e+00  2.7182818183326563803766523e+00
 28: 2.7182818233957862993910304e+00  2.7182818233958507905551755e+00
 29: 2.7182818259273639105617903e+00  2.7182818259274480086548631e+00
 30: 2.7182818271931621957604452e+00  2.7182818271932466208488932e+00
 31: 2.7182818278260525070993969e+00  2.7182818278261459278132700e+00
 32: 2.7182818278260525070993969e+00  2.7182818281425955814038786e+00
 33: 2.7182818278260525070993969e+00  2.7182818283008204083076031e+00
 34: 2.7182818278260525070993969e+00  2.7182818283799328217594654e+00
 35: 2.7182818278260525070993969e+00  2.7182818284194890285938168e+00
As you see, powl(t,N) values have saturated when n>30, where exp2l(N*log2l(t)),
which is a formula mathematically equivalent to powl(t,N), produces increasing
values in converging to Napier's constant.

The values are obtained by the program "comparePow.c":

#include <stdio.h>
#include <stdlib.h>
#include <math.h>
/*  checking the powl function.
*/

int main(int argc, char *argv[]){
  if( argc==1 ){ printf("usage: $ comparePOW n [m]"); exit(-1);}
  int n= atoi(argv[1]), m;
  if( argc==2 ) m= n; else m= atoi(argv[2]);
  for(int i= n; i<=m; i++){
    long double N=powl(2.0, (long double)i);
    if( N==N+1 ){ printf("too large N\n"); exit(-1); }
    long double t= (N+1)/N;
    long double powlx= powl(t,N);
    long double powly= exp2l(N*log2l(t));
    printf("%3d: %.25Le  %.25Le\n", i, powlx, powly);
  }

  return 0;

}

by running it with the command "./a.out 25 35".

The command "gcc -v" produces:
Configured with: --prefix=/Applications/Xcode.app/Contents/Developer/usr
--with-gxx-include-dir=/Applications/Xcode.app/Contents/Developer/Platforms/MacOSX.platform/Developer/SDKs/MacOSX10.12.sdk/usr/include/c++/4.2.1
Apple LLVM version 8.1.0 (clang-802.0.42)
Target: x86_64-apple-darwin16.7.0
Thread model: posix
InstalledDir:
/Applications/Xcode.app/Contents/Developer/Toolchains/XcodeDefault.xctoolchain/usr/bin