Average Error: 29.9 → 0.2
Time: 9.2s
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 r163705 = x;
        double r163706 = 1.0;
        double r163707 = r163705 + r163706;
        double r163708 = sqrt(r163707);
        double r163709 = sqrt(r163705);
        double r163710 = r163708 - r163709;
        return r163710;
}
↓
double f(double x) {
        double r163711 = 1.0;
        double r163712 = x;
        double r163713 = sqrt(r163712);
        double r163714 = r163712 + r163711;
        double r163715 = sqrt(r163714);
        double r163716 = r163713 + r163715;
        double r163717 = r163711 / r163716;
        return r163717;
}

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--29.8

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

  4. Simplified0.2

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

  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 2020042 +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)))