Average Error: 0.2 → 0.3
Time: 12.3s
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 r7538848 = x;
        double r7538849 = 1.0;
        double r7538850 = r7538848 + r7538849;
        double r7538851 = sqrt(r7538850);
        double r7538852 = r7538849 + r7538851;
        double r7538853 = r7538848 / r7538852;
        return r7538853;
}
↓
double f(double x) {
        double r7538854 = x;
        double r7538855 = 1.0;
        double r7538856 = r7538855 + r7538854;
        double r7538857 = sqrt(r7538856);
        double r7538858 = sqrt(r7538857);
        double r7538859 = r7538858 * r7538858;
        double r7538860 = r7538855 + r7538859;
        double r7538861 = r7538854 / r7538860;
        return r7538861;
}

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 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))