Average Error: 30.5 → 0.2
Time: 16.1s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1}{\sqrt{x} + \sqrt{x + 1}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{1}{\sqrt{x} + \sqrt{x + 1}}
double f(double x) {
        double r123416 = x;
        double r123417 = 1.0;
        double r123418 = r123416 + r123417;
        double r123419 = sqrt(r123418);
        double r123420 = sqrt(r123416);
        double r123421 = r123419 - r123420;
        return r123421;
}
↓
double f(double x) {
        double r123422 = 1.0;
        double r123423 = x;
        double r123424 = sqrt(r123423);
        double r123425 = r123423 + r123422;
        double r123426 = sqrt(r123425);
        double r123427 = r123424 + r123426;
        double r123428 = r123422 / r123427;
        return r123428;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.5
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 30.5

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

  3. Applied flip--30.3

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

  4. Simplified0.2

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

  5. Simplified0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019322 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64

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

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