Average Error: 0.2 → 0.3
Time: 4.5s
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 r171188 = x;
        double r171189 = 1.0;
        double r171190 = r171188 + r171189;
        double r171191 = sqrt(r171190);
        double r171192 = r171189 + r171191;
        double r171193 = r171188 / r171192;
        return r171193;
}
↓
double f(double x) {
        double r171194 = x;
        double r171195 = 1.0;
        double r171196 = r171194 + r171195;
        double r171197 = cbrt(r171196);
        double r171198 = fabs(r171197);
        double r171199 = sqrt(r171197);
        double r171200 = r171198 * r171199;
        double r171201 = r171195 + r171200;
        double r171202 = r171194 / r171201;
        return r171202;
}

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