Average Error: 29.6 → 0.2
Time: 19.5s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{\sqrt{\frac{1}{\sqrt{x} + \sqrt{x + 1}}} \cdot \sqrt{1}}{\sqrt{\sqrt{x} + \sqrt{x + 1}}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{\sqrt{\frac{1}{\sqrt{x} + \sqrt{x + 1}}} \cdot \sqrt{1}}{\sqrt{\sqrt{x} + \sqrt{x + 1}}}
double f(double x) {
        double r122678 = x;
        double r122679 = 1.0;
        double r122680 = r122678 + r122679;
        double r122681 = sqrt(r122680);
        double r122682 = sqrt(r122678);
        double r122683 = r122681 - r122682;
        return r122683;
}
↓
double f(double x) {
        double r122684 = 1.0;
        double r122685 = x;
        double r122686 = sqrt(r122685);
        double r122687 = r122685 + r122684;
        double r122688 = sqrt(r122687);
        double r122689 = r122686 + r122688;
        double r122690 = r122684 / r122689;
        double r122691 = sqrt(r122690);
        double r122692 = sqrt(r122684);
        double r122693 = r122691 * r122692;
        double r122694 = sqrt(r122689);
        double r122695 = r122693 / r122694;
        return r122695;
}

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

Target

Original29.6
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.6

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

  3. Applied flip--29.4

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

  4. Simplified0.2

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

  5. Simplified0.2

    \[\leadsto \frac{1}{\color{blue}{\sqrt{x} + \sqrt{x + 1}}}\]
  6. Using strategy rm

  7. Applied add-sqr-sqrt0.3

    \[\leadsto \color{blue}{\sqrt{\frac{1}{\sqrt{x} + \sqrt{x + 1}}} \cdot \sqrt{\frac{1}{\sqrt{x} + \sqrt{x + 1}}}}\]
  8. Using strategy rm

  9. Applied sqrt-div0.3

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

  10. Applied associate-*r/0.2

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

  11. Final simplification0.2

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

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64

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

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