Average Error: 29.9 → 0.2
Time: 9.0s
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 r156885 = x;
        double r156886 = 1.0;
        double r156887 = r156885 + r156886;
        double r156888 = sqrt(r156887);
        double r156889 = sqrt(r156885);
        double r156890 = r156888 - r156889;
        return r156890;
}
↓
double f(double x) {
        double r156891 = 1.0;
        double r156892 = 0.0;
        double r156893 = r156891 + r156892;
        double r156894 = x;
        double r156895 = r156894 + r156891;
        double r156896 = sqrt(r156895);
        double r156897 = sqrt(r156894);
        double r156898 = r156896 + r156897;
        double r156899 = r156893 / r156898;
        return r156899;
}

Error

Bits error versus x

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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

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2020081 +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)))