Average Error: 30.3 → 0.2
Time: 17.7s
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 r4365556 = x;
        double r4365557 = 1.0;
        double r4365558 = r4365556 + r4365557;
        double r4365559 = sqrt(r4365558);
        double r4365560 = sqrt(r4365556);
        double r4365561 = r4365559 - r4365560;
        return r4365561;
}
↓
double f(double x) {
        double r4365562 = 1.0;
        double r4365563 = x;
        double r4365564 = r4365563 + r4365562;
        double r4365565 = sqrt(r4365564);
        double r4365566 = sqrt(r4365563);
        double r4365567 = r4365565 + r4365566;
        double r4365568 = r4365562 / r4365567;
        return r4365568;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.3
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 30.3

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

  3. Applied flip--30.1

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

  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 2019171 
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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