Average Error: 29.3 → 0.2
Time: 18.3s
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 r4160809 = x;
        double r4160810 = 1.0;
        double r4160811 = r4160809 + r4160810;
        double r4160812 = sqrt(r4160811);
        double r4160813 = sqrt(r4160809);
        double r4160814 = r4160812 - r4160813;
        return r4160814;
}
↓
double f(double x) {
        double r4160815 = 1.0;
        double r4160816 = x;
        double r4160817 = r4160816 + r4160815;
        double r4160818 = sqrt(r4160817);
        double r4160819 = sqrt(r4160816);
        double r4160820 = r4160818 + r4160819;
        double r4160821 = r4160815 / r4160820;
        return r4160821;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.3
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.3

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

  3. Applied flip--29.1

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

  4. Simplified0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019158 
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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