Average Error: 30.1 → 0.2
Time: 1.4m
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 r3173771 = x;
        double r3173772 = 1.0;
        double r3173773 = r3173771 + r3173772;
        double r3173774 = sqrt(r3173773);
        double r3173775 = sqrt(r3173771);
        double r3173776 = r3173774 - r3173775;
        return r3173776;
}
↓
double f(double x) {
        double r3173777 = 1.0;
        double r3173778 = x;
        double r3173779 = r3173778 + r3173777;
        double r3173780 = sqrt(r3173779);
        double r3173781 = sqrt(r3173778);
        double r3173782 = r3173780 + r3173781;
        double r3173783 = r3173777 / r3173782;
        return r3173783;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.1
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 30.1

    \[\sqrt{x + 1} - \sqrt{x}\]
  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

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019152 
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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