Average Error: 30.8 → 0.2
Time: 4.8s
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 r486015 = x;
        double r486016 = 1.0;
        double r486017 = r486015 + r486016;
        double r486018 = sqrt(r486017);
        double r486019 = sqrt(r486015);
        double r486020 = r486018 - r486019;
        return r486020;
}
↓
double f(double x) {
        double r486021 = 1.0;
        double r486022 = x;
        double r486023 = r486022 + r486021;
        double r486024 = sqrt(r486023);
        double r486025 = sqrt(r486022);
        double r486026 = r486024 + r486025;
        double r486027 = r486021 / r486026;
        return r486027;
}

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

Target

Original30.8
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 30.8

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

  3. Applied flip--30.5

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

  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 2020002 
(FPCore (x)
  :name "Main:bigenough3 from C"
  :precision binary64

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

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