Average Error: 30.3 → 0.2
Time: 16.0s
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 r22718617 = x;
        double r22718618 = 1.0;
        double r22718619 = r22718617 + r22718618;
        double r22718620 = sqrt(r22718619);
        double r22718621 = sqrt(r22718617);
        double r22718622 = r22718620 - r22718621;
        return r22718622;
}
↓
double f(double x) {
        double r22718623 = 1.0;
        double r22718624 = x;
        double r22718625 = r22718624 + r22718623;
        double r22718626 = sqrt(r22718625);
        double r22718627 = sqrt(r22718624);
        double r22718628 = r22718626 + r22718627;
        double r22718629 = r22718623 / r22718628;
        return r22718629;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.3
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 30.3

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

  3. Applied flip--30.0

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

  4. Simplified29.6

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

  5. Taylor expanded around 0 0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019170 
(FPCore (x)
  :name "Main:bigenough3 from C"

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

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