Average Error: 19.6 → 0.4
Time: 2.6m
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\frac{1}{\sqrt{\sqrt{x + 1}} \cdot \left(\sqrt{x} \cdot \sqrt{\sqrt{x + 1}}\right)}}{\sqrt{x} + \sqrt{x + 1}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\frac{1}{\sqrt{\sqrt{x + 1}} \cdot \left(\sqrt{x} \cdot \sqrt{\sqrt{x + 1}}\right)}}{\sqrt{x} + \sqrt{x + 1}}
double f(double x) {
        double r13070610 = 1.0;
        double r13070611 = x;
        double r13070612 = sqrt(r13070611);
        double r13070613 = r13070610 / r13070612;
        double r13070614 = r13070611 + r13070610;
        double r13070615 = sqrt(r13070614);
        double r13070616 = r13070610 / r13070615;
        double r13070617 = r13070613 - r13070616;
        return r13070617;
}
↓
double f(double x) {
        double r13070618 = 1.0;
        double r13070619 = x;
        double r13070620 = r13070619 + r13070618;
        double r13070621 = sqrt(r13070620);
        double r13070622 = sqrt(r13070621);
        double r13070623 = sqrt(r13070619);
        double r13070624 = r13070623 * r13070622;
        double r13070625 = r13070622 * r13070624;
        double r13070626 = r13070618 / r13070625;
        double r13070627 = r13070623 + r13070621;
        double r13070628 = r13070626 / r13070627;
        return r13070628;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.6
Target0.6
Herbie0.4
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.6

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

  3. Applied frac-sub19.6

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

  4. Simplified19.6

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

  6. Applied flip--19.4

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

  7. Applied associate-/l/19.4

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

  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 associate-/r*0.4

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

  12. Applied add-sqr-sqrt0.4

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

  13. Applied associate-*r*0.4

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

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2019119 +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)))))