Average Error: 29.5 → 0.2
Time: 15.4s
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 r95365 = x;
        double r95366 = 1.0;
        double r95367 = r95365 + r95366;
        double r95368 = sqrt(r95367);
        double r95369 = sqrt(r95365);
        double r95370 = r95368 - r95369;
        return r95370;
}
↓
double f(double x) {
        double r95371 = 1.0;
        double r95372 = x;
        double r95373 = r95372 + r95371;
        double r95374 = sqrt(r95373);
        double r95375 = sqrt(r95372);
        double r95376 = r95374 + r95375;
        double r95377 = r95371 / r95376;
        return r95377;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.5
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.5

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

  3. Applied flip--29.2

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

  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 2019198 
(FPCore (x)
  :name "2sqrt (example 3.1)"

  :herbie-target
  (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))

  (- (sqrt (+ x 1.0)) (sqrt x)))