Average Error: 19.6 → 0.7
Time: 19.1s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x} \cdot \sqrt{x + 1}\right)\right) \cdot \frac{1}{\sqrt{x} \cdot x + \left(x + 1\right) \cdot \sqrt{x + 1}}}{\sqrt{x + 1} \cdot \sqrt{x}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x} \cdot \sqrt{x + 1}\right)\right) \cdot \frac{1}{\sqrt{x} \cdot x + \left(x + 1\right) \cdot \sqrt{x + 1}}}{\sqrt{x + 1} \cdot \sqrt{x}}
double f(double x) {
        double r4169326 = 1.0;
        double r4169327 = x;
        double r4169328 = sqrt(r4169327);
        double r4169329 = r4169326 / r4169328;
        double r4169330 = r4169327 + r4169326;
        double r4169331 = sqrt(r4169330);
        double r4169332 = r4169326 / r4169331;
        double r4169333 = r4169329 - r4169332;
        return r4169333;
}
↓
double f(double x) {
        double r4169334 = x;
        double r4169335 = 1.0;
        double r4169336 = r4169334 + r4169335;
        double r4169337 = sqrt(r4169336);
        double r4169338 = r4169337 * r4169337;
        double r4169339 = sqrt(r4169334);
        double r4169340 = r4169339 * r4169339;
        double r4169341 = r4169339 * r4169337;
        double r4169342 = r4169340 - r4169341;
        double r4169343 = r4169338 + r4169342;
        double r4169344 = r4169339 * r4169334;
        double r4169345 = r4169336 * r4169337;
        double r4169346 = r4169344 + r4169345;
        double r4169347 = r4169335 / r4169346;
        double r4169348 = r4169343 * r4169347;
        double r4169349 = r4169337 * r4169339;
        double r4169350 = r4169348 / r4169349;
        return r4169350;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.6
Target0.6
Herbie0.7
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

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. Simplified0.4

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

  9. Applied flip3-+0.8

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

  10. Applied associate-/r/0.8

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

  11. Simplified0.7

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

  12. Final simplification0.7

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

Reproduce

herbie shell --seed 2019139 
(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)))))