Average Error: 0.2 → 0.3
Time: 14.6s
Precision: 64
\[\frac{x}{1 + \sqrt{x + 1}}\]
↓
\[\frac{x}{1 + \left|\sqrt[3]{x + 1}\right| \cdot \sqrt{\sqrt[3]{\sqrt{x + 1}} \cdot \sqrt[3]{\sqrt{x + 1}}}}\]
\frac{x}{1 + \sqrt{x + 1}}
↓
\frac{x}{1 + \left|\sqrt[3]{x + 1}\right| \cdot \sqrt{\sqrt[3]{\sqrt{x + 1}} \cdot \sqrt[3]{\sqrt{x + 1}}}}
double f(double x) {
        double r195582 = x;
        double r195583 = 1.0;
        double r195584 = r195582 + r195583;
        double r195585 = sqrt(r195584);
        double r195586 = r195583 + r195585;
        double r195587 = r195582 / r195586;
        return r195587;
}
↓
double f(double x) {
        double r195588 = x;
        double r195589 = 1.0;
        double r195590 = r195588 + r195589;
        double r195591 = cbrt(r195590);
        double r195592 = fabs(r195591);
        double r195593 = sqrt(r195590);
        double r195594 = cbrt(r195593);
        double r195595 = r195594 * r195594;
        double r195596 = sqrt(r195595);
        double r195597 = r195592 * r195596;
        double r195598 = r195589 + r195597;
        double r195599 = r195588 / r195598;
        return r195599;
}

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. Using strategy rm

  7. Applied add-sqr-sqrt0.3

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

  8. Applied cbrt-prod0.3

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

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1 (sqrt (+ x 1)))))