Average Error: 30.1 → 0.2
Time: 38.1s
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 r11089296 = x;
        double r11089297 = 1.0;
        double r11089298 = r11089296 + r11089297;
        double r11089299 = sqrt(r11089298);
        double r11089300 = sqrt(r11089296);
        double r11089301 = r11089299 - r11089300;
        return r11089301;
}
↓
double f(double x) {
        double r11089302 = 1.0;
        double r11089303 = x;
        double r11089304 = r11089303 + r11089302;
        double r11089305 = sqrt(r11089304);
        double r11089306 = sqrt(r11089303);
        double r11089307 = r11089305 + r11089306;
        double r11089308 = r11089302 / r11089307;
        return r11089308;
}

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)))