Average Error: 29.5 → 0.2
Time: 1.4m
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{\left(x - x\right) + 1}{\sqrt{1 + x} + \sqrt{x}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{\left(x - x\right) + 1}{\sqrt{1 + x} + \sqrt{x}}
double f(double x) {
        double r7216831 = x;
        double r7216832 = 1.0;
        double r7216833 = r7216831 + r7216832;
        double r7216834 = sqrt(r7216833);
        double r7216835 = sqrt(r7216831);
        double r7216836 = r7216834 - r7216835;
        return r7216836;
}
↓
double f(double x) {
        double r7216837 = x;
        double r7216838 = r7216837 - r7216837;
        double r7216839 = 1.0;
        double r7216840 = r7216838 + r7216839;
        double r7216841 = r7216839 + r7216837;
        double r7216842 = sqrt(r7216841);
        double r7216843 = sqrt(r7216837);
        double r7216844 = r7216842 + r7216843;
        double r7216845 = r7216840 / r7216844;
        return r7216845;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.5
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.5

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

  3. Applied flip--29.4

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

  4. Simplified0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019200 
(FPCore (x)
  :name "2sqrt (example 3.1)"

  :herbie-target
  (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))

  (- (sqrt (+ x 1.0)) (sqrt x)))