Average Error: 29.9 → 0.2
Time: 18.0s
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 r13961586 = x;
        double r13961587 = 1.0;
        double r13961588 = r13961586 + r13961587;
        double r13961589 = sqrt(r13961588);
        double r13961590 = sqrt(r13961586);
        double r13961591 = r13961589 - r13961590;
        return r13961591;
}
↓
double f(double x) {
        double r13961592 = 1.0;
        double r13961593 = x;
        double r13961594 = r13961593 + r13961592;
        double r13961595 = sqrt(r13961594);
        double r13961596 = sqrt(r13961593);
        double r13961597 = r13961595 + r13961596;
        double r13961598 = r13961592 / r13961597;
        return r13961598;
}

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. Simplified29.2

    \[\leadsto \frac{\color{blue}{\left(x + 1\right) - x}}{\sqrt{x + 1} + \sqrt{x}}\]

  5. Taylor expanded around 0 0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019173 
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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