[BUG]: math/base/special/expm1rel overflows prematurely #12436

import expm1rel from '@stdlib/math-base-special-expm1rel';  // pnpm install @stdlib/math-base-special-expm1rel
import expm1 from '@stdlib/math-base-special-expm1';  // pnpm install @stdlib/math-base-special-expm1


// Overflow resistant expression for expm1rel(x):
//   expm1rel(x) = (exp(x)-1) / x
//               = (exp(x/2)-1) * (exp(x/2)+1) / x
//               = expm1(x/2) * (expm1(x/2) + 2) / x
//               = expm1(x/2) / x * (expm1(x/2) + 2)
const overflow_resistant_expm1rel = x => {
  if (x === 0) return 1;
  const temp = expm1(x/2);
  if (temp === Infinity) return Infinity;
  return temp / x * (temp + 2);
};
    
const data = {
  "Last value for which expm1rel correctly returns a finite number": 709.7827128933839,
  "First value for which expm1rel prematurely overflows":            709.782712893384,
  "A nice round number for which expm1rel prematurely overflows":    710,
  "Last value for which expm1rel prematurely overflows":             716.3568913878178,
  "First value for which expm1rel correctly returns Infinity":       716.3568913878179,
};
  
for (const key in data) {
  const x = data[key];
  console.log("  "+key+": "+x);
  console.log("                         expm1rel("+x+") = "+expm1rel(x));
  console.log("      overflow_resistant_expm1rel("+x+") = "+overflow_resistant_expm1rel(x));
  console.log();
}