Average Error: 30.1 → 0.2
Time: 5.3s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{\sqrt{\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}} \cdot \sqrt{1}}{\sqrt{\sqrt{x + 1} + \sqrt{x}}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{\sqrt{\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}} \cdot \sqrt{1}}{\sqrt{\sqrt{x + 1} + \sqrt{x}}}
double f(double x) {
        double r135443 = x;
        double r135444 = 1.0;
        double r135445 = r135443 + r135444;
        double r135446 = sqrt(r135445);
        double r135447 = sqrt(r135443);
        double r135448 = r135446 - r135447;
        return r135448;
}
↓
double f(double x) {
        double r135449 = 1.0;
        double r135450 = 0.0;
        double r135451 = r135449 + r135450;
        double r135452 = x;
        double r135453 = r135452 + r135449;
        double r135454 = sqrt(r135453);
        double r135455 = sqrt(r135452);
        double r135456 = r135454 + r135455;
        double r135457 = r135451 / r135456;
        double r135458 = sqrt(r135457);
        double r135459 = sqrt(r135449);
        double r135460 = r135458 * r135459;
        double r135461 = sqrt(r135456);
        double r135462 = r135460 / r135461;
        return r135462;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.1
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 30.1

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

  3. Applied flip--29.9

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

  4. Simplified0.2

    \[\leadsto \frac{\color{blue}{1 + 0}}{\sqrt{x + 1} + \sqrt{x}}\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt0.3

    \[\leadsto \color{blue}{\sqrt{\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}} \cdot \sqrt{\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}}}\]
  7. Using strategy rm

  8. Applied sqrt-div0.3

    \[\leadsto \sqrt{\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}} \cdot \color{blue}{\frac{\sqrt{1 + 0}}{\sqrt{\sqrt{x + 1} + \sqrt{x}}}}\]

  9. Applied associate-*r/0.2

    \[\leadsto \color{blue}{\frac{\sqrt{\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}} \cdot \sqrt{1 + 0}}{\sqrt{\sqrt{x + 1} + \sqrt{x}}}}\]

  10. Simplified0.2

    \[\leadsto \frac{\color{blue}{\sqrt{\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}} \cdot \sqrt{1}}}{\sqrt{\sqrt{x + 1} + \sqrt{x}}}\]

  11. Final simplification0.2

    \[\leadsto \frac{\sqrt{\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}} \cdot \sqrt{1}}{\sqrt{\sqrt{x + 1} + \sqrt{x}}}\]

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64

  :herbie-target
  (/ 1 (+ (sqrt (+ x 1)) (sqrt x)))

  (- (sqrt (+ x 1)) (sqrt x)))