Average Error: 29.4 → 0.2
Time: 16.0s
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 r2122696 = x;
        double r2122697 = 1.0;
        double r2122698 = r2122696 + r2122697;
        double r2122699 = sqrt(r2122698);
        double r2122700 = sqrt(r2122696);
        double r2122701 = r2122699 - r2122700;
        return r2122701;
}
↓
double f(double x) {
        double r2122702 = 1.0;
        double r2122703 = x;
        double r2122704 = r2122703 + r2122702;
        double r2122705 = sqrt(r2122704);
        double r2122706 = sqrt(r2122703);
        double r2122707 = r2122705 + r2122706;
        double r2122708 = r2122702 / r2122707;
        return r2122708;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.4
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.4

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

  3. Applied flip--29.2

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

  4. Simplified28.8

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

  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 2019153 
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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