Average Error: 29.9 → 0.2
Time: 22.6s
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 r7292934 = x;
        double r7292935 = 1.0;
        double r7292936 = r7292934 + r7292935;
        double r7292937 = sqrt(r7292936);
        double r7292938 = sqrt(r7292934);
        double r7292939 = r7292937 - r7292938;
        return r7292939;
}
↓
double f(double x) {
        double r7292940 = 1.0;
        double r7292941 = x;
        double r7292942 = r7292941 + r7292940;
        double r7292943 = sqrt(r7292942);
        double r7292944 = sqrt(r7292941);
        double r7292945 = r7292943 + r7292944;
        double r7292946 = r7292940 / r7292945;
        return r7292946;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.9
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.9

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

  3. Applied flip--29.7

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

  4. Taylor expanded around 0 0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019120 
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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