Average Error: 29.9 → 0.2
Time: 5.1s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{1}{\sqrt{x + 1} + \sqrt{x}}
double f(double x) {
        double r186976 = x;
        double r186977 = 1.0;
        double r186978 = r186976 + r186977;
        double r186979 = sqrt(r186978);
        double r186980 = sqrt(r186976);
        double r186981 = r186979 - r186980;
        return r186981;
}
↓
double f(double x) {
        double r186982 = 1.0;
        double r186983 = x;
        double r186984 = r186983 + r186982;
        double r186985 = sqrt(r186984);
        double r186986 = sqrt(r186983);
        double r186987 = r186985 + r186986;
        double r186988 = r186982 / r186987;
        return r186988;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.9
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.9

    \[\sqrt{x + 1} - \sqrt{x}\]
  2. Using strategy rm

  3. Applied flip--29.7

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

  4. Simplified0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019354 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64

  :herbie-target
  (/ 1 (+ (sqrt (+ x 1)) (sqrt x)))

  (- (sqrt (+ x 1)) (sqrt x)))