Average Error: 29.4 → 0.2
Time: 16.6s
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 r3314415 = x;
        double r3314416 = 1.0;
        double r3314417 = r3314415 + r3314416;
        double r3314418 = sqrt(r3314417);
        double r3314419 = sqrt(r3314415);
        double r3314420 = r3314418 - r3314419;
        return r3314420;
}
↓
double f(double x) {
        double r3314421 = 1.0;
        double r3314422 = x;
        double r3314423 = r3314422 + r3314421;
        double r3314424 = sqrt(r3314423);
        double r3314425 = sqrt(r3314422);
        double r3314426 = r3314424 + r3314425;
        double r3314427 = r3314421 / r3314426;
        return r3314427;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.4
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.4

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

  3. Applied flip--29.3

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

  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 2019146 
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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