Average Error: 29.3 → 0.2
Time: 16.4s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\sqrt{1} \cdot \frac{\sqrt{1}}{\sqrt{x + 1} + \sqrt{x}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\sqrt{1} \cdot \frac{\sqrt{1}}{\sqrt{x + 1} + \sqrt{x}}
double f(double x) {
        double r346691 = x;
        double r346692 = 1.0;
        double r346693 = r346691 + r346692;
        double r346694 = sqrt(r346693);
        double r346695 = sqrt(r346691);
        double r346696 = r346694 - r346695;
        return r346696;
}
↓
double f(double x) {
        double r346697 = 1.0;
        double r346698 = sqrt(r346697);
        double r346699 = x;
        double r346700 = r346699 + r346697;
        double r346701 = sqrt(r346700);
        double r346702 = sqrt(r346699);
        double r346703 = r346701 + r346702;
        double r346704 = r346698 / r346703;
        double r346705 = r346698 * r346704;
        return r346705;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.3
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.3

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

  3. Applied flip--29.1

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

  4. Simplified0.2

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

  6. Applied add-sqr-sqrt0.2

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

  7. Applied sqrt-prod0.2

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

  9. Applied *-un-lft-identity0.2

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

  10. Applied add-sqr-sqrt0.2

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

  11. Applied times-frac0.2

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

  12. Simplified0.2

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

  13. Simplified0.2

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

  14. Final simplification0.2

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

Reproduce

herbie shell --seed 2019326 
(FPCore (x)
  :name "Main:bigenough3 from C"
  :precision binary64

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

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