Average Error: 30.6 → 0.3
Time: 5.2s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\sqrt{{\left(\sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}\right)}^{4}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\sqrt{{\left(\sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}\right)}^{4}}
double f(double x) {
        double r128686 = x;
        double r128687 = 1.0;
        double r128688 = r128686 + r128687;
        double r128689 = sqrt(r128688);
        double r128690 = sqrt(r128686);
        double r128691 = r128689 - r128690;
        return r128691;
}
↓
double f(double x) {
        double r128692 = 1.0;
        double r128693 = x;
        double r128694 = r128693 + r128692;
        double r128695 = sqrt(r128694);
        double r128696 = sqrt(r128693);
        double r128697 = r128695 + r128696;
        double r128698 = r128692 / r128697;
        double r128699 = sqrt(r128698);
        double r128700 = 4.0;
        double r128701 = pow(r128699, r128700);
        double r128702 = sqrt(r128701);
        return r128702;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.6
Target0.2
Herbie0.3
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 30.6

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

  3. Applied flip--30.4

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

  4. Simplified0.2

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

  6. Applied add-sqr-sqrt0.3

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

  8. Applied sqrt-div0.3

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

  10. Applied sqrt-undiv0.3

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

  11. Applied sqrt-unprod0.2

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

  12. Simplified0.3

    \[\leadsto \sqrt{\color{blue}{{\left(\sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}\right)}^{4}}}\]

  13. Final simplification0.3

    \[\leadsto \sqrt{{\left(\sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}\right)}^{4}}\]

Reproduce

herbie shell --seed 2020025 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64

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

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