Average Error: 29.5 → 0.3
Time: 11.5s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\sqrt{\frac{1 \cdot 1}{\left(\sqrt{x} + \sqrt{x + 1}\right) \cdot \left(\sqrt{x} + \sqrt{x + 1}\right)}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\sqrt{\frac{1 \cdot 1}{\left(\sqrt{x} + \sqrt{x + 1}\right) \cdot \left(\sqrt{x} + \sqrt{x + 1}\right)}}
double f(double x) {
        double r96985 = x;
        double r96986 = 1.0;
        double r96987 = r96985 + r96986;
        double r96988 = sqrt(r96987);
        double r96989 = sqrt(r96985);
        double r96990 = r96988 - r96989;
        return r96990;
}
↓
double f(double x) {
        double r96991 = 1.0;
        double r96992 = r96991 * r96991;
        double r96993 = x;
        double r96994 = sqrt(r96993);
        double r96995 = r96993 + r96991;
        double r96996 = sqrt(r96995);
        double r96997 = r96994 + r96996;
        double r96998 = r96997 * r96997;
        double r96999 = r96992 / r96998;
        double r97000 = sqrt(r96999);
        return r97000;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.5
Target0.2
Herbie0.3
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.5

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

  3. Applied flip--29.4

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

  4. Simplified0.2

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

  5. Simplified0.2

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

  7. Applied add-sqr-sqrt0.3

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

  8. Simplified0.3

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

  9. Simplified0.3

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

  11. Applied sqrt-unprod0.2

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

  12. Simplified0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2020046 +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)))