Average Error: 19.8 → 0.4
Time: 13.7s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\left(1 \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\left(1 \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}}
double f(double x) {
        double r157825 = 1.0;
        double r157826 = x;
        double r157827 = sqrt(r157826);
        double r157828 = r157825 / r157827;
        double r157829 = r157826 + r157825;
        double r157830 = sqrt(r157829);
        double r157831 = r157825 / r157830;
        double r157832 = r157828 - r157831;
        return r157832;
}
↓
double f(double x) {
        double r157833 = 1.0;
        double r157834 = x;
        double r157835 = r157834 + r157833;
        double r157836 = sqrt(r157835);
        double r157837 = sqrt(r157834);
        double r157838 = r157836 + r157837;
        double r157839 = r157833 / r157838;
        double r157840 = r157833 * r157839;
        double r157841 = 1.0;
        double r157842 = r157837 * r157836;
        double r157843 = r157841 / r157842;
        double r157844 = r157840 * r157843;
        return r157844;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.8
Target0.6
Herbie0.4
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

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

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

  6. Applied flip--19.5

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

  7. Simplified19.1

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

  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 div-inv0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019198 
(FPCore (x)
  :name "2isqrt (example 3.6)"

  :herbie-target
  (/ 1.0 (+ (* (+ x 1.0) (sqrt x)) (* x (sqrt (+ x 1.0)))))

  (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))