Average Error: 29.5 → 0.2
Time: 14.9s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[{\left(\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)\right)}^{\left(-\frac{1}{2}\right)}\]
\sqrt{x + 1} - \sqrt{x}
↓
{\left(\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)\right)}^{\left(-\frac{1}{2}\right)}
double f(double x) {
        double r2877874 = x;
        double r2877875 = 1.0;
        double r2877876 = r2877874 + r2877875;
        double r2877877 = sqrt(r2877876);
        double r2877878 = sqrt(r2877874);
        double r2877879 = r2877877 - r2877878;
        return r2877879;
}
↓
double f(double x) {
        double r2877880 = 1.0;
        double r2877881 = x;
        double r2877882 = r2877880 + r2877881;
        double r2877883 = sqrt(r2877882);
        double r2877884 = sqrt(r2877881);
        double r2877885 = r2877883 + r2877884;
        double r2877886 = r2877885 * r2877885;
        double r2877887 = 0.5;
        double r2877888 = -r2877887;
        double r2877889 = pow(r2877886, r2877888);
        return r2877889;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 29.5

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

  3. Applied flip--29.3

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

  4. Simplified0.2

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

  6. Applied add-sqr-sqrt0.3

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

  8. Applied pow1/20.3

    \[\leadsto \frac{1}{\sqrt{\sqrt{x + 1} + \sqrt{x}} \cdot \color{blue}{{\left(\sqrt{x + 1} + \sqrt{x}\right)}^{\frac{1}{2}}}}\]

  9. Applied pow1/20.3

    \[\leadsto \frac{1}{\color{blue}{{\left(\sqrt{x + 1} + \sqrt{x}\right)}^{\frac{1}{2}}} \cdot {\left(\sqrt{x + 1} + \sqrt{x}\right)}^{\frac{1}{2}}}\]

  10. Applied pow-prod-down0.2

    \[\leadsto \frac{1}{\color{blue}{{\left(\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)\right)}^{\frac{1}{2}}}}\]

  11. Applied pow-flip0.2

    \[\leadsto \color{blue}{{\left(\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)\right)}^{\left(-\frac{1}{2}\right)}}\]

  12. Final simplification0.2

    \[\leadsto {\left(\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)\right)}^{\left(-\frac{1}{2}\right)}\]

Reproduce

herbie shell --seed 2019134 
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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