Average Error: 20.3 → 0.4
Time: 32.3s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\frac{1}{\sqrt{x}}}{\frac{\sqrt{x + 1} + \sqrt{x}}{\frac{1}{\sqrt{x + 1}}}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\frac{1}{\sqrt{x}}}{\frac{\sqrt{x + 1} + \sqrt{x}}{\frac{1}{\sqrt{x + 1}}}}
double f(double x) {
        double r10108864 = 1.0;
        double r10108865 = x;
        double r10108866 = sqrt(r10108865);
        double r10108867 = r10108864 / r10108866;
        double r10108868 = r10108865 + r10108864;
        double r10108869 = sqrt(r10108868);
        double r10108870 = r10108864 / r10108869;
        double r10108871 = r10108867 - r10108870;
        return r10108871;
}
↓
double f(double x) {
        double r10108872 = 1.0;
        double r10108873 = x;
        double r10108874 = sqrt(r10108873);
        double r10108875 = r10108872 / r10108874;
        double r10108876 = r10108873 + r10108872;
        double r10108877 = sqrt(r10108876);
        double r10108878 = r10108877 + r10108874;
        double r10108879 = r10108872 / r10108877;
        double r10108880 = r10108878 / r10108879;
        double r10108881 = r10108875 / r10108880;
        return r10108881;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.3
Target0.6
Herbie0.4
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 20.3

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

  3. Applied frac-sub20.3

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

  4. Simplified20.3

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

  6. Applied flip--20.0

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

  7. Applied associate-/l/20.0

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

  8. Simplified0.8

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

  10. Applied associate-/r*0.4

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

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

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

  13. Applied times-frac0.4

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

  14. Applied associate-/l*0.4

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

  15. Final simplification0.4

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

Reproduce

herbie shell --seed 2019124 
(FPCore (x)
  :name "2isqrt (example 3.6)"

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

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