Average Error: 30.2 → 0.2
Time: 10.6s
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 r623336 = x;
        double r623337 = 1.0;
        double r623338 = r623336 + r623337;
        double r623339 = sqrt(r623338);
        double r623340 = sqrt(r623336);
        double r623341 = r623339 - r623340;
        return r623341;
}
↓
double f(double x) {
        double r623342 = 1.0;
        double r623343 = x;
        double r623344 = r623343 + r623342;
        double r623345 = sqrt(r623344);
        double r623346 = sqrt(r623343);
        double r623347 = r623345 + r623346;
        double r623348 = r623342 / r623347;
        return r623348;
}

Error

Bits error versus x

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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--29.9

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

  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 2020047 
(FPCore (x)
  :name "Main:bigenough3 from C"
  :precision binary64

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

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