Average Error: 29.7 → 0.2
Time: 18.9s
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 r113278 = x;
        double r113279 = 1.0;
        double r113280 = r113278 + r113279;
        double r113281 = sqrt(r113280);
        double r113282 = sqrt(r113278);
        double r113283 = r113281 - r113282;
        return r113283;
}
↓
double f(double x) {
        double r113284 = 1.0;
        double r113285 = x;
        double r113286 = r113285 + r113284;
        double r113287 = sqrt(r113286);
        double r113288 = sqrt(r113285);
        double r113289 = r113287 + r113288;
        double r113290 = r113284 / r113289;
        return r113290;
}

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--29.6

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

  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 2019195 
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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