Average Error: 30.1 → 0.2
Time: 19.0s
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 r1552624 = x;
        double r1552625 = 1.0;
        double r1552626 = r1552624 + r1552625;
        double r1552627 = sqrt(r1552626);
        double r1552628 = sqrt(r1552624);
        double r1552629 = r1552627 - r1552628;
        return r1552629;
}
↓
double f(double x) {
        double r1552630 = 1.0;
        double r1552631 = x;
        double r1552632 = r1552631 + r1552630;
        double r1552633 = sqrt(r1552632);
        double r1552634 = sqrt(r1552631);
        double r1552635 = r1552633 + r1552634;
        double r1552636 = r1552630 / r1552635;
        return r1552636;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 30.1

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

  3. Applied flip--29.8

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

  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 2019152 
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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