Average Error: 30.1 → 0.2
Time: 1.2m
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 r2651739 = x;
        double r2651740 = 1.0;
        double r2651741 = r2651739 + r2651740;
        double r2651742 = sqrt(r2651741);
        double r2651743 = sqrt(r2651739);
        double r2651744 = r2651742 - r2651743;
        return r2651744;
}
↓
double f(double x) {
        double r2651745 = 1.0;
        double r2651746 = x;
        double r2651747 = r2651746 + r2651745;
        double r2651748 = sqrt(r2651747);
        double r2651749 = sqrt(r2651746);
        double r2651750 = r2651748 + r2651749;
        double r2651751 = r2651745 / r2651750;
        return r2651751;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 30.1

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

  3. Applied flip--29.8

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

  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 2019152 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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