Average Error: 19.9 → 0.5
Time: 13.4s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{1 \cdot \frac{1}{\sqrt{x} + \sqrt{x + 1}}}{\left(\left|\sqrt[3]{x + 1}\right| \cdot \sqrt{x}\right) \cdot \sqrt{\sqrt[3]{x + 1}}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{1 \cdot \frac{1}{\sqrt{x} + \sqrt{x + 1}}}{\left(\left|\sqrt[3]{x + 1}\right| \cdot \sqrt{x}\right) \cdot \sqrt{\sqrt[3]{x + 1}}}
double f(double x) {
        double r116661 = 1.0;
        double r116662 = x;
        double r116663 = sqrt(r116662);
        double r116664 = r116661 / r116663;
        double r116665 = r116662 + r116661;
        double r116666 = sqrt(r116665);
        double r116667 = r116661 / r116666;
        double r116668 = r116664 - r116667;
        return r116668;
}
↓
double f(double x) {
        double r116669 = 1.0;
        double r116670 = x;
        double r116671 = sqrt(r116670);
        double r116672 = r116670 + r116669;
        double r116673 = sqrt(r116672);
        double r116674 = r116671 + r116673;
        double r116675 = r116669 / r116674;
        double r116676 = r116669 * r116675;
        double r116677 = cbrt(r116672);
        double r116678 = fabs(r116677);
        double r116679 = r116678 * r116671;
        double r116680 = sqrt(r116677);
        double r116681 = r116679 * r116680;
        double r116682 = r116676 / r116681;
        return r116682;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.9
Target0.7
Herbie0.5
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.9

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

  3. Applied frac-sub19.9

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

  4. Simplified19.9

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

  6. Applied flip--19.7

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

  7. Simplified19.3

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

  8. Simplified19.3

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

  9. Taylor expanded around 0 0.4

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

  11. Applied add-cube-cbrt0.5

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

  12. Applied sqrt-prod0.5

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

  13. Applied associate-*r*0.5

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

  14. Simplified0.5

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

  15. Final simplification0.5

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

Reproduce

herbie shell --seed 2019208 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64

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

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