Average Error: 0.2 → 0.3
Time: 14.4s
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 r196414 = x;
        double r196415 = 1.0;
        double r196416 = r196414 + r196415;
        double r196417 = sqrt(r196416);
        double r196418 = r196415 + r196417;
        double r196419 = r196414 / r196418;
        return r196419;
}
↓
double f(double x) {
        double r196420 = x;
        double r196421 = 1.0;
        double r196422 = r196420 + r196421;
        double r196423 = cbrt(r196422);
        double r196424 = fabs(r196423);
        double r196425 = sqrt(r196422);
        double r196426 = cbrt(r196425);
        double r196427 = r196426 * r196426;
        double r196428 = sqrt(r196427);
        double r196429 = r196424 * r196428;
        double r196430 = r196421 + r196429;
        double r196431 = r196420 / r196430;
        return r196431;
}

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)))))