Average Error: 29.5 → 0.2
Time: 15.6s
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 r412960 = x;
        double r412961 = 1.0;
        double r412962 = r412960 + r412961;
        double r412963 = sqrt(r412962);
        double r412964 = sqrt(r412960);
        double r412965 = r412963 - r412964;
        return r412965;
}
↓
double f(double x) {
        double r412966 = 1.0;
        double r412967 = x;
        double r412968 = r412967 + r412966;
        double r412969 = sqrt(r412968);
        double r412970 = sqrt(r412967);
        double r412971 = r412969 + r412970;
        double r412972 = r412966 / r412971;
        return r412972;
}

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

Target

Original29.5
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.5

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

  3. Applied flip--29.2

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

  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 2019198 
(FPCore (x)
  :name "Main:bigenough3 from C"

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

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