Average Error: 0.2 → 0.2
Time: 8.9s
Precision: 64
\[\frac{x}{1 + \sqrt{x + 1}}\]
↓
\[x \cdot \frac{1}{1 + \sqrt{x + 1}}\]
\frac{x}{1 + \sqrt{x + 1}}
↓
x \cdot \frac{1}{1 + \sqrt{x + 1}}
double f(double x) {
        double r106612 = x;
        double r106613 = 1.0;
        double r106614 = r106612 + r106613;
        double r106615 = sqrt(r106614);
        double r106616 = r106613 + r106615;
        double r106617 = r106612 / r106616;
        return r106617;
}
↓
double f(double x) {
        double r106618 = x;
        double r106619 = 1.0;
        double r106620 = 1.0;
        double r106621 = r106618 + r106620;
        double r106622 = sqrt(r106621);
        double r106623 = r106620 + r106622;
        double r106624 = r106619 / r106623;
        double r106625 = r106618 * r106624;
        return r106625;
}

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 div-inv0.2

    \[\leadsto \color{blue}{x \cdot \frac{1}{1 + \sqrt{1 + x}}}\]

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019174 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))