Average Error: 29.9 → 0.2
Time: 15.8s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1}{\sqrt{x} + \sqrt{x + 1}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{1}{\sqrt{x} + \sqrt{x + 1}}
double f(double x) {
        double r85143 = x;
        double r85144 = 1.0;
        double r85145 = r85143 + r85144;
        double r85146 = sqrt(r85145);
        double r85147 = sqrt(r85143);
        double r85148 = r85146 - r85147;
        return r85148;
}
↓
double f(double x) {
        double r85149 = 1.0;
        double r85150 = x;
        double r85151 = sqrt(r85150);
        double r85152 = r85150 + r85149;
        double r85153 = sqrt(r85152);
        double r85154 = r85151 + r85153;
        double r85155 = r85149 / r85154;
        return r85155;
}

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. Simplified0.2

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

  5. Simplified0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019304 +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)))