Average Error: 0.2 → 0.2
Time: 10.5s
Precision: 64
\[\frac{x}{1 + \sqrt{x + 1}}\]
↓
\[\frac{x}{1 + \sqrt{1 + x}}\]
\frac{x}{1 + \sqrt{x + 1}}
↓
\frac{x}{1 + \sqrt{1 + x}}
double f(double x) {
        double r101859 = x;
        double r101860 = 1.0;
        double r101861 = r101859 + r101860;
        double r101862 = sqrt(r101861);
        double r101863 = r101860 + r101862;
        double r101864 = r101859 / r101863;
        return r101864;
}
↓
double f(double x) {
        double r101865 = x;
        double r101866 = 1.0;
        double r101867 = r101866 + r101865;
        double r101868 = sqrt(r101867);
        double r101869 = r101866 + r101868;
        double r101870 = r101865 / r101869;
        return r101870;
}

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.2

    \[\frac{x}{1 + \sqrt{x + 1}}\]

  2. Simplified0.2

    \[\leadsto \color{blue}{\frac{x}{1 + \sqrt{1 + x}}}\]
  3. Using strategy rm

  4. Applied pow10.2

    \[\leadsto \frac{x}{1 + \color{blue}{{\left(\sqrt{1 + x}\right)}^{1}}}\]

  5. Final simplification0.2

    \[\leadsto \frac{x}{1 + \sqrt{1 + x}}\]

Reproduce

herbie shell --seed 2019179 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))