Average Error: 20.0 → 0.6
Time: 15.8s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{1}{\frac{\frac{\sqrt{x + 1} \cdot x + \left(x + 1\right) \cdot \sqrt{x}}{1}}{1}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{1}{\frac{\frac{\sqrt{x + 1} \cdot x + \left(x + 1\right) \cdot \sqrt{x}}{1}}{1}}
double f(double x) {
        double r133733 = 1.0;
        double r133734 = x;
        double r133735 = sqrt(r133734);
        double r133736 = r133733 / r133735;
        double r133737 = r133734 + r133733;
        double r133738 = sqrt(r133737);
        double r133739 = r133733 / r133738;
        double r133740 = r133736 - r133739;
        return r133740;
}
↓
double f(double x) {
        double r133741 = 1.0;
        double r133742 = x;
        double r133743 = 1.0;
        double r133744 = r133742 + r133743;
        double r133745 = sqrt(r133744);
        double r133746 = r133745 * r133742;
        double r133747 = sqrt(r133742);
        double r133748 = r133744 * r133747;
        double r133749 = r133746 + r133748;
        double r133750 = r133749 / r133743;
        double r133751 = r133750 / r133743;
        double r133752 = r133741 / r133751;
        return r133752;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.0
Target0.6
Herbie0.6
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 20.0

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

  3. Applied frac-sub20.0

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

  4. Simplified20.0

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

  6. Applied flip--19.8

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

  7. Simplified19.4

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

  8. Taylor expanded around 0 0.4

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

  10. Applied clear-num0.8

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

  11. Simplified0.6

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

  12. Final simplification0.6

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

Reproduce

herbie shell --seed 2019306 
(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)))))