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 r88081 = x;
        double r88082 = 1.0;
        double r88083 = r88081 + r88082;
        double r88084 = sqrt(r88083);
        double r88085 = r88082 + r88084;
        double r88086 = r88081 / r88085;
        return r88086;
}
↓
double f(double x) {
        double r88087 = x;
        double r88088 = 1.0;
        double r88089 = r88087 + r88088;
        double r88090 = sqrt(r88089);
        double r88091 = sqrt(r88090);
        double r88092 = r88091 * r88091;
        double r88093 = r88088 + r88092;
        double r88094 = r88087 / r88093;
        return r88094;
}

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 2019362 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1 (sqrt (+ x 1)))))