Average Error: 29.9 → 0.2
Time: 1.4m
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 r2501970 = x;
        double r2501971 = 1.0;
        double r2501972 = r2501970 + r2501971;
        double r2501973 = sqrt(r2501972);
        double r2501974 = sqrt(r2501970);
        double r2501975 = r2501973 - r2501974;
        return r2501975;
}
↓
double f(double x) {
        double r2501976 = 1.0;
        double r2501977 = x;
        double r2501978 = r2501977 + r2501976;
        double r2501979 = sqrt(r2501978);
        double r2501980 = sqrt(r2501977);
        double r2501981 = r2501979 + r2501980;
        double r2501982 = r2501976 / r2501981;
        return r2501982;
}

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

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019151 
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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