Average Error: 30.0 → 0.2
Time: 24.3s
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 r12162772 = x;
        double r12162773 = 1.0;
        double r12162774 = r12162772 + r12162773;
        double r12162775 = sqrt(r12162774);
        double r12162776 = sqrt(r12162772);
        double r12162777 = r12162775 - r12162776;
        return r12162777;
}
↓
double f(double x) {
        double r12162778 = 1.0;
        double r12162779 = x;
        double r12162780 = r12162779 + r12162778;
        double r12162781 = sqrt(r12162780);
        double r12162782 = sqrt(r12162779);
        double r12162783 = r12162781 + r12162782;
        double r12162784 = r12162778 / r12162783;
        return r12162784;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.0
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 30.0

    \[\sqrt{x + 1} - \sqrt{x}\]
  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. Taylor expanded around 0 0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019124 
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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