Average Error: 29.7 → 0.2
Time: 38.6s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[{\left(\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)\right)}^{\frac{-1}{2}}\]
\sqrt{x + 1} - \sqrt{x}
↓
{\left(\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)\right)}^{\frac{-1}{2}}
double f(double x) {
        double r12596875 = x;
        double r12596876 = 1.0;
        double r12596877 = r12596875 + r12596876;
        double r12596878 = sqrt(r12596877);
        double r12596879 = sqrt(r12596875);
        double r12596880 = r12596878 - r12596879;
        return r12596880;
}
↓
double f(double x) {
        double r12596881 = 1.0;
        double r12596882 = x;
        double r12596883 = r12596881 + r12596882;
        double r12596884 = sqrt(r12596883);
        double r12596885 = sqrt(r12596882);
        double r12596886 = r12596884 + r12596885;
        double r12596887 = r12596886 * r12596886;
        double r12596888 = -0.5;
        double r12596889 = pow(r12596887, r12596888);
        return r12596889;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.7
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.7

    \[\sqrt{x + 1} - \sqrt{x}\]
  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

    \[\leadsto \frac{\color{blue}{1}}{\sqrt{x + 1} + \sqrt{x}}\]
  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. Simplified0.2

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

  13. Final simplification0.2

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

Reproduce

herbie shell --seed 2019128 
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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