Average Error: 30.1 → 0.2
Time: 15.8s
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 r126586 = x;
        double r126587 = 1.0;
        double r126588 = r126586 + r126587;
        double r126589 = sqrt(r126588);
        double r126590 = sqrt(r126586);
        double r126591 = r126589 - r126590;
        return r126591;
}
↓
double f(double x) {
        double r126592 = 1.0;
        double r126593 = x;
        double r126594 = r126593 + r126592;
        double r126595 = sqrt(r126594);
        double r126596 = sqrt(r126593);
        double r126597 = r126595 + r126596;
        double r126598 = r126592 / r126597;
        return r126598;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.1
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 30.1

    \[\sqrt{x + 1} - \sqrt{x}\]

  2. Simplified30.1

    \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}}\]
  3. Using strategy rm

  4. Applied flip--29.8

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

  5. Simplified0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019174 
(FPCore (x)
  :name "2sqrt (example 3.1)"

  :herbie-target
  (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))

  (- (sqrt (+ x 1.0)) (sqrt x)))