Average Error: 30.3 → 0.2
Time: 5.1s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}
double f(double x) {
        double r178910 = x;
        double r178911 = 1.0;
        double r178912 = r178910 + r178911;
        double r178913 = sqrt(r178912);
        double r178914 = sqrt(r178910);
        double r178915 = r178913 - r178914;
        return r178915;
}
↓
double f(double x) {
        double r178916 = 1.0;
        double r178917 = 0.0;
        double r178918 = r178916 + r178917;
        double r178919 = x;
        double r178920 = r178919 + r178916;
        double r178921 = sqrt(r178920);
        double r178922 = sqrt(r178919);
        double r178923 = r178921 + r178922;
        double r178924 = r178918 / r178923;
        return r178924;
}

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

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019344 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64

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

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