Average Error: 30.1 → 0.2
Time: 13.8s
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 r5681600 = x;
        double r5681601 = 1.0;
        double r5681602 = r5681600 + r5681601;
        double r5681603 = sqrt(r5681602);
        double r5681604 = sqrt(r5681600);
        double r5681605 = r5681603 - r5681604;
        return r5681605;
}
↓
double f(double x) {
        double r5681606 = 1.0;
        double r5681607 = x;
        double r5681608 = r5681607 + r5681606;
        double r5681609 = sqrt(r5681608);
        double r5681610 = sqrt(r5681607);
        double r5681611 = r5681609 + r5681610;
        double r5681612 = r5681606 / r5681611;
        return r5681612;
}

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--29.8

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

  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 2019174 
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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