Average Error: 30.1 → 0.2
Time: 5.7s
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 r122625 = x;
        double r122626 = 1.0;
        double r122627 = r122625 + r122626;
        double r122628 = sqrt(r122627);
        double r122629 = sqrt(r122625);
        double r122630 = r122628 - r122629;
        return r122630;
}
↓
double f(double x) {
        double r122631 = 1.0;
        double r122632 = x;
        double r122633 = r122632 + r122631;
        double r122634 = sqrt(r122633);
        double r122635 = sqrt(r122632);
        double r122636 = r122634 + r122635;
        double r122637 = r122631 / r122636;
        return r122637;
}

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. Using strategy rm

  3. Applied flip--29.9

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

  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 2019353 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64

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

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