Average Error: 30.1 → 0.2
Time: 12.6s
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 r5141868 = x;
        double r5141869 = 1.0;
        double r5141870 = r5141868 + r5141869;
        double r5141871 = sqrt(r5141870);
        double r5141872 = sqrt(r5141868);
        double r5141873 = r5141871 - r5141872;
        return r5141873;
}
↓
double f(double x) {
        double r5141874 = 1.0;
        double r5141875 = x;
        double r5141876 = r5141875 + r5141874;
        double r5141877 = sqrt(r5141876);
        double r5141878 = sqrt(r5141875);
        double r5141879 = r5141877 + r5141878;
        double r5141880 = r5141874 / r5141879;
        return r5141880;
}

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

    \[\leadsto \frac{\color{blue}{1 + 0}}{\sqrt{x + 1} + \sqrt{x}}\]
  5. Using strategy rm

  6. Applied *-un-lft-identity0.2

    \[\leadsto \frac{1 + 0}{\color{blue}{1 \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}}\]

  7. Applied associate-/r*0.2

    \[\leadsto \color{blue}{\frac{\frac{1 + 0}{1}}{\sqrt{x + 1} + \sqrt{x}}}\]

  8. Simplified0.2

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

  9. Final simplification0.2

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

Reproduce

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

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

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