Average Error: 30.2 → 0.2
Time: 18.7s
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 r3155521 = x;
        double r3155522 = 1.0;
        double r3155523 = r3155521 + r3155522;
        double r3155524 = sqrt(r3155523);
        double r3155525 = sqrt(r3155521);
        double r3155526 = r3155524 - r3155525;
        return r3155526;
}
↓
double f(double x) {
        double r3155527 = 1.0;
        double r3155528 = x;
        double r3155529 = r3155528 + r3155527;
        double r3155530 = sqrt(r3155529);
        double r3155531 = sqrt(r3155528);
        double r3155532 = r3155530 + r3155531;
        double r3155533 = r3155527 / r3155532;
        return r3155533;
}

Error

Bits error versus x

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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

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019135 
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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