Average Error: 29.1 → 0.2
Time: 40.5s
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 r14156796 = x;
        double r14156797 = 1.0;
        double r14156798 = r14156796 + r14156797;
        double r14156799 = sqrt(r14156798);
        double r14156800 = sqrt(r14156796);
        double r14156801 = r14156799 - r14156800;
        return r14156801;
}
↓
double f(double x) {
        double r14156802 = 1.0;
        double r14156803 = x;
        double r14156804 = r14156802 + r14156803;
        double r14156805 = sqrt(r14156804);
        double r14156806 = sqrt(r14156803);
        double r14156807 = r14156805 + r14156806;
        double r14156808 = r14156807 * r14156807;
        double r14156809 = -0.5;
        double r14156810 = pow(r14156808, r14156809);
        return r14156810;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 29.1

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

  3. Applied flip--28.9

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

  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 inv-pow0.3

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

  9. Applied sqrt-pow10.3

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

  10. Applied inv-pow0.3

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

  11. Applied sqrt-pow10.3

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

  12. Applied pow-prod-down0.2

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

  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 2019125 
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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