Average Error: 19.8 → 0.7
Time: 17.4s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x} \cdot \sqrt{x + 1}\right)\right) \cdot \frac{1}{\sqrt{x} \cdot x + \left(x + 1\right) \cdot \sqrt{x + 1}}}{\sqrt{x + 1} \cdot \sqrt{x}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x} \cdot \sqrt{x + 1}\right)\right) \cdot \frac{1}{\sqrt{x} \cdot x + \left(x + 1\right) \cdot \sqrt{x + 1}}}{\sqrt{x + 1} \cdot \sqrt{x}}
double f(double x) {
        double r2644752 = 1.0;
        double r2644753 = x;
        double r2644754 = sqrt(r2644753);
        double r2644755 = r2644752 / r2644754;
        double r2644756 = r2644753 + r2644752;
        double r2644757 = sqrt(r2644756);
        double r2644758 = r2644752 / r2644757;
        double r2644759 = r2644755 - r2644758;
        return r2644759;
}
↓
double f(double x) {
        double r2644760 = x;
        double r2644761 = 1.0;
        double r2644762 = r2644760 + r2644761;
        double r2644763 = sqrt(r2644762);
        double r2644764 = r2644763 * r2644763;
        double r2644765 = sqrt(r2644760);
        double r2644766 = r2644765 * r2644765;
        double r2644767 = r2644765 * r2644763;
        double r2644768 = r2644766 - r2644767;
        double r2644769 = r2644764 + r2644768;
        double r2644770 = r2644765 * r2644760;
        double r2644771 = r2644762 * r2644763;
        double r2644772 = r2644770 + r2644771;
        double r2644773 = r2644761 / r2644772;
        double r2644774 = r2644769 * r2644773;
        double r2644775 = r2644763 * r2644765;
        double r2644776 = r2644774 / r2644775;
        return r2644776;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.8
Target0.7
Herbie0.7
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.8

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

  3. Applied frac-sub19.8

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

  4. Simplified19.8

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

  6. Applied flip--19.5

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

  7. Simplified0.4

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

  9. Applied flip3-+0.8

    \[\leadsto \frac{\frac{1}{\color{blue}{\frac{{\left(\sqrt{x + 1}\right)}^{3} + {\left(\sqrt{x}\right)}^{3}}{\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)}}}}{\sqrt{x} \cdot \sqrt{x + 1}}\]

  10. Applied associate-/r/0.8

    \[\leadsto \frac{\color{blue}{\frac{1}{{\left(\sqrt{x + 1}\right)}^{3} + {\left(\sqrt{x}\right)}^{3}} \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right)}}{\sqrt{x} \cdot \sqrt{x + 1}}\]

  11. Simplified0.7

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

  12. Final simplification0.7

    \[\leadsto \frac{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x} \cdot \sqrt{x + 1}\right)\right) \cdot \frac{1}{\sqrt{x} \cdot x + \left(x + 1\right) \cdot \sqrt{x + 1}}}{\sqrt{x + 1} \cdot \sqrt{x}}\]

Reproduce

herbie shell --seed 2019137 
(FPCore (x)
  :name "2isqrt (example 3.6)"

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

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