Average Error: 29.7 → 0.2
Time: 14.6s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{\sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}} \cdot \sqrt{1}}{\sqrt{\sqrt{x + 1} + \sqrt{x}}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{\sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}} \cdot \sqrt{1}}{\sqrt{\sqrt{x + 1} + \sqrt{x}}}
double f(double x) {
        double r384434 = x;
        double r384435 = 1.0;
        double r384436 = r384434 + r384435;
        double r384437 = sqrt(r384436);
        double r384438 = sqrt(r384434);
        double r384439 = r384437 - r384438;
        return r384439;
}
↓
double f(double x) {
        double r384440 = 1.0;
        double r384441 = x;
        double r384442 = r384441 + r384440;
        double r384443 = sqrt(r384442);
        double r384444 = sqrt(r384441);
        double r384445 = r384443 + r384444;
        double r384446 = r384440 / r384445;
        double r384447 = sqrt(r384446);
        double r384448 = sqrt(r384440);
        double r384449 = r384447 * r384448;
        double r384450 = sqrt(r384445);
        double r384451 = r384449 / r384450;
        return r384451;
}

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--29.5

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

  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 sqrt-div0.3

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

  9. Applied associate-*r/0.2

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2019212 
(FPCore (x)
  :name "Main:bigenough3 from C"
  :precision binary64

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

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