Average Error: 30.1 → 0.2
Time: 13.2s
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 r343692 = x;
        double r343693 = 1.0;
        double r343694 = r343692 + r343693;
        double r343695 = sqrt(r343694);
        double r343696 = sqrt(r343692);
        double r343697 = r343695 - r343696;
        return r343697;
}
↓
double f(double x) {
        double r343698 = 1.0;
        double r343699 = x;
        double r343700 = r343699 + r343698;
        double r343701 = sqrt(r343700);
        double r343702 = sqrt(r343699);
        double r343703 = r343701 + r343702;
        double r343704 = r343698 / r343703;
        return r343704;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

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. Simplified30.1

    \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}}\]
  3. Using strategy rm

  4. Applied flip--29.8

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

  5. Simplified0.2

    \[\leadsto \frac{\color{blue}{1 + 0}}{\sqrt{1 + x} + \sqrt{x}}\]
  6. Using strategy rm

  7. Applied pow10.2

    \[\leadsto \frac{1 + 0}{\color{blue}{{\left(\sqrt{1 + x} + \sqrt{x}\right)}^{1}}}\]

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x)
  :name "Main:bigenough3 from C"

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

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