Average Error: 0.2 → 0.2
Time: 3.4s
Precision: 64
\[\frac{x}{1 + \sqrt{x + 1}}\]
↓
\[\frac{x}{1 + \sqrt{\sqrt{x + 1}} \cdot \sqrt{\sqrt{x + 1}}}\]
\frac{x}{1 + \sqrt{x + 1}}
↓
\frac{x}{1 + \sqrt{\sqrt{x + 1}} \cdot \sqrt{\sqrt{x + 1}}}
double f(double x) {
        double r110654 = x;
        double r110655 = 1.0;
        double r110656 = r110654 + r110655;
        double r110657 = sqrt(r110656);
        double r110658 = r110655 + r110657;
        double r110659 = r110654 / r110658;
        return r110659;
}
↓
double f(double x) {
        double r110660 = x;
        double r110661 = 1.0;
        double r110662 = r110660 + r110661;
        double r110663 = sqrt(r110662);
        double r110664 = sqrt(r110663);
        double r110665 = r110664 * r110664;
        double r110666 = r110661 + r110665;
        double r110667 = r110660 / r110666;
        return r110667;
}

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-sqr-sqrt0.2

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

  4. Applied sqrt-prod0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2020027 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1 (sqrt (+ x 1)))))