Average Error: 29.7 → 0.2
Time: 17.2s
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 r3402839 = x;
        double r3402840 = 1.0;
        double r3402841 = r3402839 + r3402840;
        double r3402842 = sqrt(r3402841);
        double r3402843 = sqrt(r3402839);
        double r3402844 = r3402842 - r3402843;
        return r3402844;
}
↓
double f(double x) {
        double r3402845 = 1.0;
        double r3402846 = x;
        double r3402847 = r3402846 + r3402845;
        double r3402848 = sqrt(r3402847);
        double r3402849 = sqrt(r3402846);
        double r3402850 = r3402848 + r3402849;
        double r3402851 = r3402845 / r3402850;
        return r3402851;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.7
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.7

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

  3. Applied flip--29.5

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

  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 2019144 
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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