Average Error: 30.0 → 0.2
Time: 16.2s
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 r4007599 = x;
        double r4007600 = 1.0;
        double r4007601 = r4007599 + r4007600;
        double r4007602 = sqrt(r4007601);
        double r4007603 = sqrt(r4007599);
        double r4007604 = r4007602 - r4007603;
        return r4007604;
}
↓
double f(double x) {
        double r4007605 = 1.0;
        double r4007606 = x;
        double r4007607 = r4007606 + r4007605;
        double r4007608 = sqrt(r4007607);
        double r4007609 = sqrt(r4007606);
        double r4007610 = r4007608 + r4007609;
        double r4007611 = r4007605 / r4007610;
        return r4007611;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.0
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 30.0

    \[\sqrt{x + 1} - \sqrt{x}\]
  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 2019141 
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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