Average Error: 30.6 → 0.3
Time: 7.9s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\sqrt{{\left(\sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}\right)}^{4}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\sqrt{{\left(\sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}\right)}^{4}}
double f(double x) {
        double r465 = x;
        double r466 = 1.0;
        double r467 = r465 + r466;
        double r468 = sqrt(r467);
        double r469 = sqrt(r465);
        double r470 = r468 - r469;
        return r470;
}
↓
double f(double x) {
        double r471 = 1.0;
        double r472 = x;
        double r473 = r472 + r471;
        double r474 = sqrt(r473);
        double r475 = sqrt(r472);
        double r476 = r474 + r475;
        double r477 = r471 / r476;
        double r478 = sqrt(r477);
        double r479 = 4.0;
        double r480 = pow(r478, r479);
        double r481 = sqrt(r480);
        return r481;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.6
Target0.2
Herbie0.3
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 30.6

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

  3. Applied flip--30.4

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

  4. Simplified0.2

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

  6. Applied add-sqr-sqrt0.3

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

  8. Applied sqrt-unprod0.2

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

  9. Simplified0.3

    \[\leadsto \sqrt{\color{blue}{{\left(\sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}\right)}^{4}}}\]

  10. Final simplification0.3

    \[\leadsto \sqrt{{\left(\sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}\right)}^{4}}\]

Reproduce

herbie shell --seed 2020025 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64

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

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