Average Error: 29.9 → 0.2
Time: 18.0s
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 r3686851 = x;
        double r3686852 = 1.0;
        double r3686853 = r3686851 + r3686852;
        double r3686854 = sqrt(r3686853);
        double r3686855 = sqrt(r3686851);
        double r3686856 = r3686854 - r3686855;
        return r3686856;
}
↓
double f(double x) {
        double r3686857 = 1.0;
        double r3686858 = x;
        double r3686859 = r3686858 + r3686857;
        double r3686860 = sqrt(r3686859);
        double r3686861 = sqrt(r3686858);
        double r3686862 = r3686860 + r3686861;
        double r3686863 = r3686857 / r3686862;
        return r3686863;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.9
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.9

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

  3. Applied flip--29.6

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

  4. Simplified29.2

    \[\leadsto \frac{\color{blue}{\left(1 + x\right) - x}}{\sqrt{x + 1} + \sqrt{x}}\]

  5. Simplified29.2

    \[\leadsto \frac{\left(1 + x\right) - x}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}}\]
  6. Using strategy rm

  7. Applied clear-num29.2

    \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{x} + \sqrt{1 + x}}{\left(1 + x\right) - x}}}\]

  8. Simplified0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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