Average Error: 29.8 → 0.2
Time: 23.1s
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 r8785625 = x;
        double r8785626 = 1.0;
        double r8785627 = r8785625 + r8785626;
        double r8785628 = sqrt(r8785627);
        double r8785629 = sqrt(r8785625);
        double r8785630 = r8785628 - r8785629;
        return r8785630;
}
↓
double f(double x) {
        double r8785631 = 1.0;
        double r8785632 = x;
        double r8785633 = r8785632 + r8785631;
        double r8785634 = sqrt(r8785633);
        double r8785635 = sqrt(r8785632);
        double r8785636 = r8785634 + r8785635;
        double r8785637 = r8785631 / r8785636;
        return r8785637;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.8
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.8

    \[\sqrt{x + 1} - \sqrt{x}\]
  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. Taylor expanded around 0 0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019119 
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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