Average Error: 30.1 → 0.3
Time: 19.6s
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 r123690 = x;
        double r123691 = 1.0;
        double r123692 = r123690 + r123691;
        double r123693 = sqrt(r123692);
        double r123694 = sqrt(r123690);
        double r123695 = r123693 - r123694;
        return r123695;
}
↓
double f(double x) {
        double r123696 = 1.0;
        double r123697 = x;
        double r123698 = r123697 + r123696;
        double r123699 = sqrt(r123698);
        double r123700 = sqrt(r123697);
        double r123701 = r123699 + r123700;
        double r123702 = r123696 / r123701;
        double r123703 = sqrt(r123702);
        double r123704 = 1.0;
        double r123705 = sqrt(r123701);
        double r123706 = r123704 / r123705;
        double r123707 = r123696 / r123705;
        double r123708 = r123706 * r123707;
        double r123709 = sqrt(r123708);
        double r123710 = r123703 * r123709;
        return r123710;
}

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 "2sqrt (example 3.1)"

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

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