Average Error: 0.2 → 0.3
Time: 13.1s
Precision: 64
\[\frac{x}{1 + \sqrt{x + 1}}\]
↓
\[\frac{x}{1 + \sqrt{\sqrt{1 + x}} \cdot \sqrt{\sqrt{1 + x}}}\]
\frac{x}{1 + \sqrt{x + 1}}
↓
\frac{x}{1 + \sqrt{\sqrt{1 + x}} \cdot \sqrt{\sqrt{1 + x}}}
double f(double x) {
        double r4885035 = x;
        double r4885036 = 1.0;
        double r4885037 = r4885035 + r4885036;
        double r4885038 = sqrt(r4885037);
        double r4885039 = r4885036 + r4885038;
        double r4885040 = r4885035 / r4885039;
        return r4885040;
}
↓
double f(double x) {
        double r4885041 = x;
        double r4885042 = 1.0;
        double r4885043 = r4885042 + r4885041;
        double r4885044 = sqrt(r4885043);
        double r4885045 = sqrt(r4885044);
        double r4885046 = r4885045 * r4885045;
        double r4885047 = r4885042 + r4885046;
        double r4885048 = r4885041 / r4885047;
        return r4885048;
}

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-prod0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019192 +o rules:numerics
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))