Average Error: 20.0 → 0.7
Time: 5.7s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{1 \cdot 1}{\sqrt{x} \cdot \left(\sqrt{x + 1} \cdot \sqrt{x} + \left(x + 1\right)\right)}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{1 \cdot 1}{\sqrt{x} \cdot \left(\sqrt{x + 1} \cdot \sqrt{x} + \left(x + 1\right)\right)}
double f(double x) {
        double r172450 = 1.0;
        double r172451 = x;
        double r172452 = sqrt(r172451);
        double r172453 = r172450 / r172452;
        double r172454 = r172451 + r172450;
        double r172455 = sqrt(r172454);
        double r172456 = r172450 / r172455;
        double r172457 = r172453 - r172456;
        return r172457;
}
↓
double f(double x) {
        double r172458 = 1.0;
        double r172459 = r172458 * r172458;
        double r172460 = x;
        double r172461 = sqrt(r172460);
        double r172462 = r172460 + r172458;
        double r172463 = sqrt(r172462);
        double r172464 = r172463 * r172461;
        double r172465 = r172464 + r172462;
        double r172466 = r172461 * r172465;
        double r172467 = r172459 / r172466;
        return r172467;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.0
Target0.6
Herbie0.7
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

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. Simplified19.3

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

  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 associate-*r/0.4

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

  11. Applied associate-/l/0.8

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

  12. Simplified0.7

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

  13. Final simplification0.7

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

Reproduce

herbie shell --seed 2020021 
(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)))))