Average Error: 19.8 → 0.5
Time: 2.5m
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{1}{\left(\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt{x}\right) \cdot \sqrt{\sqrt[3]{x + 1}}} \cdot \frac{1}{\sqrt{x} + \sqrt{x + 1}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{1}{\left(\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt{x}\right) \cdot \sqrt{\sqrt[3]{x + 1}}} \cdot \frac{1}{\sqrt{x} + \sqrt{x + 1}}
double f(double x) {
        double r11962772 = 1.0;
        double r11962773 = x;
        double r11962774 = sqrt(r11962773);
        double r11962775 = r11962772 / r11962774;
        double r11962776 = r11962773 + r11962772;
        double r11962777 = sqrt(r11962776);
        double r11962778 = r11962772 / r11962777;
        double r11962779 = r11962775 - r11962778;
        return r11962779;
}
↓
double f(double x) {
        double r11962780 = 1.0;
        double r11962781 = x;
        double r11962782 = r11962781 + r11962780;
        double r11962783 = cbrt(r11962782);
        double r11962784 = r11962783 * r11962783;
        double r11962785 = sqrt(r11962784);
        double r11962786 = sqrt(r11962781);
        double r11962787 = r11962785 * r11962786;
        double r11962788 = sqrt(r11962783);
        double r11962789 = r11962787 * r11962788;
        double r11962790 = r11962780 / r11962789;
        double r11962791 = sqrt(r11962782);
        double r11962792 = r11962786 + r11962791;
        double r11962793 = r11962780 / r11962792;
        double r11962794 = r11962790 * r11962793;
        return r11962794;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.8
Target0.6
Herbie0.5
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.8

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

  3. Applied frac-sub19.7

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

  4. Simplified19.7

    \[\leadsto \frac{\color{blue}{\sqrt{x + 1} - \sqrt{x}}}{\sqrt{x} \cdot \sqrt{x + 1}}\]
  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. Applied associate-/l/19.5

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

  8. Simplified0.8

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

  10. Applied *-un-lft-identity0.8

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

  11. Applied times-frac0.4

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

  13. Applied add-cube-cbrt0.5

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

  14. Applied sqrt-prod0.5

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

  15. Applied associate-*r*0.5

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

  16. Final simplification0.5

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

Reproduce

herbie shell --seed 2019120 +o rules:numerics
(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)))))