Average Error: 30.6 → 0.2
Time: 12.9s
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 r366645 = x;
        double r366646 = 1.0;
        double r366647 = r366645 + r366646;
        double r366648 = sqrt(r366647);
        double r366649 = sqrt(r366645);
        double r366650 = r366648 - r366649;
        return r366650;
}
↓
double f(double x) {
        double r366651 = 1.0;
        double r366652 = x;
        double r366653 = r366652 + r366651;
        double r366654 = sqrt(r366653);
        double r366655 = sqrt(r366652);
        double r366656 = r366654 + r366655;
        double r366657 = r366651 / r366656;
        return r366657;
}

Error

Bits error versus x

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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

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019209 
(FPCore (x)
  :name "Main:bigenough3 from C"
  :precision binary64

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

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