Average Error: 19.7 → 0.5
Time: 16.0s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{1 \cdot \left(\frac{1}{\sqrt{\sqrt{x} + \sqrt{x + 1}}} \cdot \frac{1}{\sqrt{\sqrt{x} + \sqrt{x + 1}}}\right)}{\sqrt{x} \cdot \sqrt{x + 1}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{1 \cdot \left(\frac{1}{\sqrt{\sqrt{x} + \sqrt{x + 1}}} \cdot \frac{1}{\sqrt{\sqrt{x} + \sqrt{x + 1}}}\right)}{\sqrt{x} \cdot \sqrt{x + 1}}
double f(double x) {
        double r127897 = 1.0;
        double r127898 = x;
        double r127899 = sqrt(r127898);
        double r127900 = r127897 / r127899;
        double r127901 = r127898 + r127897;
        double r127902 = sqrt(r127901);
        double r127903 = r127897 / r127902;
        double r127904 = r127900 - r127903;
        return r127904;
}
↓
double f(double x) {
        double r127905 = 1.0;
        double r127906 = 1.0;
        double r127907 = x;
        double r127908 = sqrt(r127907);
        double r127909 = r127907 + r127905;
        double r127910 = sqrt(r127909);
        double r127911 = r127908 + r127910;
        double r127912 = sqrt(r127911);
        double r127913 = r127906 / r127912;
        double r127914 = r127905 / r127912;
        double r127915 = r127913 * r127914;
        double r127916 = r127905 * r127915;
        double r127917 = r127908 * r127910;
        double r127918 = r127916 / r127917;
        return r127918;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.7
Target0.7
Herbie0.5
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.7

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

  3. Applied frac-sub19.7

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

  4. Simplified19.7

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

  6. Applied flip--19.5

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

  7. Simplified19.0

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

  8. Simplified19.0

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

  9. Taylor expanded around 0 0.4

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

  11. Applied add-sqr-sqrt0.4

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

  12. Applied *-un-lft-identity0.4

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

  13. Applied times-frac0.5

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

  14. Final simplification0.5

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

Reproduce

herbie shell --seed 2019235 
(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)))))