Average Error: 30.1 → 0.3
Time: 21.0s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}} \cdot \sqrt{\frac{1}{\sqrt{\sqrt{x + 1} + \sqrt{x}}} \cdot \frac{1}{\sqrt{\sqrt{x + 1} + \sqrt{x}}}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}} \cdot \sqrt{\frac{1}{\sqrt{\sqrt{x + 1} + \sqrt{x}}} \cdot \frac{1}{\sqrt{\sqrt{x + 1} + \sqrt{x}}}}
double f(double x) {
        double r326895 = x;
        double r326896 = 1.0;
        double r326897 = r326895 + r326896;
        double r326898 = sqrt(r326897);
        double r326899 = sqrt(r326895);
        double r326900 = r326898 - r326899;
        return r326900;
}
↓
double f(double x) {
        double r326901 = 1.0;
        double r326902 = x;
        double r326903 = r326902 + r326901;
        double r326904 = sqrt(r326903);
        double r326905 = sqrt(r326902);
        double r326906 = r326904 + r326905;
        double r326907 = r326901 / r326906;
        double r326908 = sqrt(r326907);
        double r326909 = 1.0;
        double r326910 = sqrt(r326906);
        double r326911 = r326909 / r326910;
        double r326912 = r326901 / r326910;
        double r326913 = r326911 * r326912;
        double r326914 = sqrt(r326913);
        double r326915 = r326908 * r326914;
        return r326915;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.1
Target0.2
Herbie0.3
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

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

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

  6. Applied add-sqr-sqrt0.3

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

  8. Applied add-sqr-sqrt0.4

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

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

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

  10. Applied times-frac0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2019199 
(FPCore (x)
  :name "Main:bigenough3 from C"

  :herbie-target
  (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))

  (- (sqrt (+ x 1.0)) (sqrt x)))