Average Error: 29.9 → 0.2
Time: 29.5s
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 r7334321 = x;
        double r7334322 = 1.0;
        double r7334323 = r7334321 + r7334322;
        double r7334324 = sqrt(r7334323);
        double r7334325 = sqrt(r7334321);
        double r7334326 = r7334324 - r7334325;
        return r7334326;
}
↓
double f(double x) {
        double r7334327 = 1.0;
        double r7334328 = x;
        double r7334329 = r7334328 + r7334327;
        double r7334330 = sqrt(r7334329);
        double r7334331 = sqrt(r7334328);
        double r7334332 = r7334330 + r7334331;
        double r7334333 = r7334327 / r7334332;
        return r7334333;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.9
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.9

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

  3. Applied flip--29.7

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

  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 2019121 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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