Average Error: 29.5 → 0.2
Time: 15.8s
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 r3986579 = x;
        double r3986580 = 1.0;
        double r3986581 = r3986579 + r3986580;
        double r3986582 = sqrt(r3986581);
        double r3986583 = sqrt(r3986579);
        double r3986584 = r3986582 - r3986583;
        return r3986584;
}
↓
double f(double x) {
        double r3986585 = 1.0;
        double r3986586 = x;
        double r3986587 = r3986586 + r3986585;
        double r3986588 = sqrt(r3986587);
        double r3986589 = sqrt(r3986586);
        double r3986590 = r3986588 + r3986589;
        double r3986591 = r3986585 / r3986590;
        return r3986591;
}

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--29.3

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

  4. Simplified0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019162 
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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