Average Error: 30.3 → 0.2
Time: 14.3s
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 r5902627 = x;
        double r5902628 = 1.0;
        double r5902629 = r5902627 + r5902628;
        double r5902630 = sqrt(r5902629);
        double r5902631 = sqrt(r5902627);
        double r5902632 = r5902630 - r5902631;
        return r5902632;
}
↓
double f(double x) {
        double r5902633 = 1.0;
        double r5902634 = x;
        double r5902635 = r5902634 + r5902633;
        double r5902636 = sqrt(r5902635);
        double r5902637 = sqrt(r5902634);
        double r5902638 = r5902636 + r5902637;
        double r5902639 = r5902633 / r5902638;
        return r5902639;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.3
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 30.3

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

  3. Applied flip--30.1

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

  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 2019165 
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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