Average Error: 29.6 → 0.2
Time: 19.9s
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 r95858 = x;
        double r95859 = 1.0;
        double r95860 = r95858 + r95859;
        double r95861 = sqrt(r95860);
        double r95862 = sqrt(r95858);
        double r95863 = r95861 - r95862;
        return r95863;
}
↓
double f(double x) {
        double r95864 = 1.0;
        double r95865 = x;
        double r95866 = sqrt(r95865);
        double r95867 = r95865 + r95864;
        double r95868 = sqrt(r95867);
        double r95869 = r95866 + r95868;
        double r95870 = r95864 / r95869;
        return r95870;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.6
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.6

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

  3. Applied flip--29.4

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

  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 2019323 +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)))