Average Error: 29.0 → 0.3
Time: 17.9s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1}{\sqrt{\sqrt{x + 1} + \sqrt{x}}} \cdot \sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{1}{\sqrt{\sqrt{x + 1} + \sqrt{x}}} \cdot \sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}
double f(double x) {
        double r2165401 = x;
        double r2165402 = 1.0;
        double r2165403 = r2165401 + r2165402;
        double r2165404 = sqrt(r2165403);
        double r2165405 = sqrt(r2165401);
        double r2165406 = r2165404 - r2165405;
        return r2165406;
}
↓
double f(double x) {
        double r2165407 = 1.0;
        double r2165408 = x;
        double r2165409 = r2165408 + r2165407;
        double r2165410 = sqrt(r2165409);
        double r2165411 = sqrt(r2165408);
        double r2165412 = r2165410 + r2165411;
        double r2165413 = sqrt(r2165412);
        double r2165414 = r2165407 / r2165413;
        double r2165415 = r2165407 / r2165412;
        double r2165416 = sqrt(r2165415);
        double r2165417 = r2165414 * r2165416;
        return r2165417;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.0
Target0.2
Herbie0.3
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.0

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

  3. Applied flip--28.8

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

  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. Simplified0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2019130 
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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