Average Error: 30.2 → 0.2
Time: 5.2s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}
double f(double x) {
        double r140224 = x;
        double r140225 = 1.0;
        double r140226 = r140224 + r140225;
        double r140227 = sqrt(r140226);
        double r140228 = sqrt(r140224);
        double r140229 = r140227 - r140228;
        return r140229;
}
↓
double f(double x) {
        double r140230 = 1.0;
        double r140231 = 0.0;
        double r140232 = r140230 + r140231;
        double r140233 = x;
        double r140234 = r140233 + r140230;
        double r140235 = sqrt(r140234);
        double r140236 = sqrt(r140233);
        double r140237 = r140235 + r140236;
        double r140238 = r140232 / r140237;
        return r140238;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 30.2

    \[\sqrt{x + 1} - \sqrt{x}\]
  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. Simplified0.2

    \[\leadsto \frac{\color{blue}{1 + 0}}{\sqrt{x + 1} + \sqrt{x}}\]

  5. Final simplification0.2

    \[\leadsto \frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}\]

Reproduce

herbie shell --seed 2019352 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64

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

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