Average Error: 30.1 → 0.2
Time: 1.5m
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 r37040055 = x;
        double r37040056 = 1.0;
        double r37040057 = r37040055 + r37040056;
        double r37040058 = sqrt(r37040057);
        double r37040059 = sqrt(r37040055);
        double r37040060 = r37040058 - r37040059;
        return r37040060;
}
↓
double f(double x) {
        double r37040061 = 1.0;
        double r37040062 = x;
        double r37040063 = r37040062 + r37040061;
        double r37040064 = sqrt(r37040063);
        double r37040065 = sqrt(r37040062);
        double r37040066 = r37040064 + r37040065;
        double r37040067 = r37040061 / r37040066;
        return r37040067;
}

Error

Bits error versus x

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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 2019174 
(FPCore (x)
  :name "Main:bigenough3 from C"

  :herbie-target
  (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))

  (- (sqrt (+ x 1.0)) (sqrt x)))