Average Error: 30.6 → 0.2
Time: 13.5s
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 r126388 = x;
        double r126389 = 1.0;
        double r126390 = r126388 + r126389;
        double r126391 = sqrt(r126390);
        double r126392 = sqrt(r126388);
        double r126393 = r126391 - r126392;
        return r126393;
}
↓
double f(double x) {
        double r126394 = 1.0;
        double r126395 = x;
        double r126396 = sqrt(r126395);
        double r126397 = r126395 + r126394;
        double r126398 = sqrt(r126397);
        double r126399 = r126396 + r126398;
        double r126400 = r126394 / r126399;
        return r126400;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

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

    \[\leadsto \frac{\color{blue}{1 + 0}}{\sqrt{x + 1} + \sqrt{x}}\]

  5. Simplified0.2

    \[\leadsto \frac{1 + 0}{\color{blue}{\sqrt{x} + \sqrt{x + 1}}}\]

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019209 +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)))