Average Error: 30.4 → 0.2
Time: 16.0s
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 r3896959 = x;
        double r3896960 = 1.0;
        double r3896961 = r3896959 + r3896960;
        double r3896962 = sqrt(r3896961);
        double r3896963 = sqrt(r3896959);
        double r3896964 = r3896962 - r3896963;
        return r3896964;
}
↓
double f(double x) {
        double r3896965 = 1.0;
        double r3896966 = x;
        double r3896967 = r3896966 + r3896965;
        double r3896968 = sqrt(r3896967);
        double r3896969 = sqrt(r3896966);
        double r3896970 = r3896968 + r3896969;
        double r3896971 = r3896965 / r3896970;
        return r3896971;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.4
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 30.4

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

  3. Applied flip--30.2

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

  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 2019129 
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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