Average Error: 29.7 → 0.2
Time: 24.9s
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 r3264334 = x;
        double r3264335 = 1.0;
        double r3264336 = r3264334 + r3264335;
        double r3264337 = sqrt(r3264336);
        double r3264338 = sqrt(r3264334);
        double r3264339 = r3264337 - r3264338;
        return r3264339;
}
↓
double f(double x) {
        double r3264340 = 1.0;
        double r3264341 = x;
        double r3264342 = r3264341 + r3264340;
        double r3264343 = sqrt(r3264342);
        double r3264344 = sqrt(r3264341);
        double r3264345 = r3264343 + r3264344;
        double r3264346 = r3264340 / r3264345;
        return r3264346;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.7
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.7

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

  3. Applied flip--29.5

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

  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 2019139 
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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