Average Error: 29.9 → 0.2
Time: 18.2s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{1}{\sqrt{x + 1} + \sqrt{x}}
double f(double x) {
        double r3877089 = x;
        double r3877090 = 1.0;
        double r3877091 = r3877089 + r3877090;
        double r3877092 = sqrt(r3877091);
        double r3877093 = sqrt(r3877089);
        double r3877094 = r3877092 - r3877093;
        return r3877094;
}
↓
double f(double x) {
        double r3877095 = 1.0;
        double r3877096 = x;
        double r3877097 = r3877096 + r3877095;
        double r3877098 = sqrt(r3877097);
        double r3877099 = sqrt(r3877096);
        double r3877100 = r3877098 + r3877099;
        double r3877101 = r3877095 / r3877100;
        return r3877101;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.9
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.9

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

  3. Applied flip--29.6

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

  4. Simplified29.2

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

  5. Simplified29.2

    \[\leadsto \frac{\left(1 + x\right) - x}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}}\]
  6. Using strategy rm

  7. Applied clear-num29.2

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

  8. Simplified0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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