Average Error: 0.2 → 0.3
Time: 3.9s
Precision: 64
\[\frac{x}{1 + \sqrt{x + 1}}\]
↓
\[\frac{x}{1 + \left|\sqrt[3]{x + 1}\right| \cdot \sqrt{\sqrt[3]{x + 1}}}\]
\frac{x}{1 + \sqrt{x + 1}}
↓
\frac{x}{1 + \left|\sqrt[3]{x + 1}\right| \cdot \sqrt{\sqrt[3]{x + 1}}}
double f(double x) {
        double r75718 = x;
        double r75719 = 1.0;
        double r75720 = r75718 + r75719;
        double r75721 = sqrt(r75720);
        double r75722 = r75719 + r75721;
        double r75723 = r75718 / r75722;
        return r75723;
}
↓
double f(double x) {
        double r75724 = x;
        double r75725 = 1.0;
        double r75726 = r75724 + r75725;
        double r75727 = cbrt(r75726);
        double r75728 = fabs(r75727);
        double r75729 = sqrt(r75727);
        double r75730 = r75728 * r75729;
        double r75731 = r75725 + r75730;
        double r75732 = r75724 / r75731;
        return r75732;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.2

    \[\frac{x}{1 + \sqrt{x + 1}}\]
  2. Using strategy rm

  3. Applied add-cube-cbrt0.3

    \[\leadsto \frac{x}{1 + \sqrt{\color{blue}{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}}}}\]

  4. Applied sqrt-prod0.3

    \[\leadsto \frac{x}{1 + \color{blue}{\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt{\sqrt[3]{x + 1}}}}\]

  5. Simplified0.3

    \[\leadsto \frac{x}{1 + \color{blue}{\left|\sqrt[3]{x + 1}\right|} \cdot \sqrt{\sqrt[3]{x + 1}}}\]

  6. Final simplification0.3

    \[\leadsto \frac{x}{1 + \left|\sqrt[3]{x + 1}\right| \cdot \sqrt{\sqrt[3]{x + 1}}}\]

Reproduce

herbie shell --seed 2020034 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1 (sqrt (+ x 1)))))