Average Error: 0.2 → 0.3
Time: 12.5s
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 r95507 = x;
        double r95508 = 1.0;
        double r95509 = r95507 + r95508;
        double r95510 = sqrt(r95509);
        double r95511 = r95508 + r95510;
        double r95512 = r95507 / r95511;
        return r95512;
}
↓
double f(double x) {
        double r95513 = x;
        double r95514 = 1.0;
        double r95515 = r95513 + r95514;
        double r95516 = cbrt(r95515);
        double r95517 = fabs(r95516);
        double r95518 = sqrt(r95515);
        double r95519 = cbrt(r95518);
        double r95520 = r95519 * r95519;
        double r95521 = sqrt(r95520);
        double r95522 = r95517 * r95521;
        double r95523 = r95514 + r95522;
        double r95524 = r95513 / r95523;
        return r95524;
}

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