Average Error: 19.6 → 0.6
Time: 16.0s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\left(1 + \left(x - x\right)\right) \cdot 1}{\sqrt{x} \cdot \left(x + 1\right) + \sqrt{x + 1} \cdot x}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\left(1 + \left(x - x\right)\right) \cdot 1}{\sqrt{x} \cdot \left(x + 1\right) + \sqrt{x + 1} \cdot x}
double f(double x) {
        double r113433 = 1.0;
        double r113434 = x;
        double r113435 = sqrt(r113434);
        double r113436 = r113433 / r113435;
        double r113437 = r113434 + r113433;
        double r113438 = sqrt(r113437);
        double r113439 = r113433 / r113438;
        double r113440 = r113436 - r113439;
        return r113440;
}
↓
double f(double x) {
        double r113441 = 1.0;
        double r113442 = x;
        double r113443 = r113442 - r113442;
        double r113444 = r113441 + r113443;
        double r113445 = r113444 * r113441;
        double r113446 = sqrt(r113442);
        double r113447 = r113442 + r113441;
        double r113448 = r113446 * r113447;
        double r113449 = sqrt(r113447);
        double r113450 = r113449 * r113442;
        double r113451 = r113448 + r113450;
        double r113452 = r113445 / r113451;
        return r113452;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

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

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

  5. Simplified19.6

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

  7. Applied flip--19.4

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

  8. Simplified0.4

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

  10. Applied associate-*r/0.4

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

  11. Applied associate-/l/0.7

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

  12. Simplified0.6

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

  13. Final simplification0.6

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

Reproduce

herbie shell --seed 2019179 
(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)))))