Average Error: 30.2 → 0.2
Time: 5.3s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}
double f(double x) {
        double r582784 = x;
        double r582785 = 1.0;
        double r582786 = r582784 + r582785;
        double r582787 = sqrt(r582786);
        double r582788 = sqrt(r582784);
        double r582789 = r582787 - r582788;
        return r582789;
}
↓
double f(double x) {
        double r582790 = 1.0;
        double r582791 = 0.0;
        double r582792 = r582790 + r582791;
        double r582793 = x;
        double r582794 = r582793 + r582790;
        double r582795 = sqrt(r582794);
        double r582796 = sqrt(r582793);
        double r582797 = r582795 + r582796;
        double r582798 = r582792 / r582797;
        return r582798;
}

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 30.2

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

  3. Applied flip--30.0

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

  4. Simplified0.2

    \[\leadsto \frac{\color{blue}{1 + 0}}{\sqrt{x + 1} + \sqrt{x}}\]

  5. Final simplification0.2

    \[\leadsto \frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}\]

Reproduce

herbie shell --seed 2020083 +o rules:numerics
(FPCore (x)
  :name "Main:bigenough3 from C"
  :precision binary64

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

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