Average Error: 29.6 → 0.2
Time: 15.7s
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 r4529503 = x;
        double r4529504 = 1.0;
        double r4529505 = r4529503 + r4529504;
        double r4529506 = sqrt(r4529505);
        double r4529507 = sqrt(r4529503);
        double r4529508 = r4529506 - r4529507;
        return r4529508;
}
↓
double f(double x) {
        double r4529509 = 1.0;
        double r4529510 = x;
        double r4529511 = r4529510 + r4529509;
        double r4529512 = sqrt(r4529511);
        double r4529513 = sqrt(r4529510);
        double r4529514 = r4529512 + r4529513;
        double r4529515 = r4529509 / r4529514;
        return r4529515;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.6
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.6

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

  3. Applied flip--29.4

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

  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 2019172 
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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